Amongst the four common methods available for determining the surface tension of liquids and solutions (ring method, plate method, pendant drop, and bubble pressure method) it worthwhile to note some differences in capabilities and limitations. In particular, because the most widely used of these methods, historically, has been the ring method. And, it is also, by far also the most limited and the most readily riddled by potential errors, and even false surface tension readings, for certain surfactant solutions. This is a common issue in industrial surface science. Therefore, the intent of this paper is to describe the methods, and point out the possible pitfalls of ring method tensiometry.

Ring Method

The ring method has three main issues, which make it a less than good option for measuring exacting surface tensions, particularly on surfactant based solutions, wherein the rate of surfactant diffusion to new formed surfaces is particularly slow, as is the case particularly for large molecule surfactants, amphoterics, and fluorosurfactants.

1. Most importantly, the ring method itself is designed to keep the surface in a nonequilibrium state during the measurement of surface tension. The ring is pulled through the surface to make the measurement (or in today’s available high end tensiometers, at minimum, the ring expands and contracts the surface during the measurement – looking for the maximum force of the liquid meniscus). So the measurement of surface tension is really made on a surface which is in a nonequilibrium state. This does not matter to the measurement of surface tension, if you are measuring a pure liquid. Because for pure liquids, the surface tension is, at all times, the same. However, in surfactant solutions, wherein the surface tension is dependent on the presence and orientation of the surfactants at the surface, having the surface in a state of expansion during the measurement can make a huge difference in the measured surface tension. As an example, witness the data below, all performed with an, as nearly as possible, true and round DuNouy ring, and with the necessary Harkins and Jordan correction factors for the mass of liquid trapped under the ring not due to surface tension (two other issues we will discuss shortly). The only difference between these measurements of surface tension is the speed at which the ring is being pulled through the surface – as noted. However, two different liquids are tested. One is pure water, the other a solution of a simple non-ionic surfactant (nonylphenol ethoxylate with an average degree of ethoxylation of 9.5 units in water at 100 mg/L).

Ring Pull-Through Rate During Surface Tension Measurement (mm/min)	Surface Tension Measured (corrected) Pure Water (mN/m)	Surface Tension Measured (corrected) Nonyl-Phenol Ethoxylate Solution (mN/m)
10	72.51	36.34
5	72.50	35.24
2	72.53	34.02
1	72.54	33.19
0.5	72.50	32.34

Note that the measured surface tension changes with the rate of ring pull through, in the case of the surfactant solution, but not in the case of the pure liquid. The actual equilibrium surface tension for this surfactant solution is 30.12 mN/m – which we establish shortly. But, regardless of that, this is what is deceiving about using the ring method for equilibrium surface tension measurement.

Most people use pure liquids (water, typically) as justification that the surface tension measurements they are making on unknown solutions are accurate. They then ignore (or have no true control over, in the case of manual ring tensiometers) the rate of ring pull through.

However, in a surfactant system, the true equilibrium surface tension is dependent on how the molecules of the surfactant are adsorb at, and orienting at, the surface. This requires time, and doesn’t ever truly happen if the surface is being stretched (more surface created) during measurement of the surface tension. The problem can be minimized somewhat by reducing the rate at which the ring is pulled through the surface. However, a true equilibrium tension is never measured by the ring method. The surface tension measured is always somewhat higher than equilibrium – and the extent to which it is higher is based on rate of pull through, as well as the slowness of the surfactant equilibration process. Particularly difficult applications for measurement include large molecule surfactants, amphoterics, and fluorosurfactants – known to be notoriously slow to reach final equilibration at surface.

2. I also mentioned briefly above a correction factor was necessary whenever surface tension is measured by the ring method. This is to account for the fact that the ring pulls a meniscus above the surface of liquid during measurement. The portion of the liquid pulled above the surface, which is directly under the ring, is
not there due to surface tension forces – but rather capillary forces. See schematic below. But, this liquid does contribute force to the force measuring device used to measure surface tension in a ring method experiment. So, the resultant surface tension (or the force value from which it comes) needs to be corrected for that extra force, in order to measure a true surface tension. The amount of that force varies with meniscus height, which, in turn varies with surface tension and density of the liquid being measured for surface tension. However, a typical error in surface tension, due to not making the proper correction for this effect is about,
7% (causing a further increase in the reported surface tension).

3. The third problem with the ring method is quite simply that the rings are difficult to keep true – circular and free of bending to the shaft – which causes the ring to be not absolutely parallel with the surface, as the meniscus is pulled. This also typically results in more force being generated (higher than expected surface
tensions measured) – and/or causes the meniscus to tear before the measurement can be made.

Plate Method

Another means of measuring “equilibrium” surface tension is the Wilhelmy plate method. This method is similar to the ring method, except that the plate is a flat piece of platinum, instead of a ring, and that a meniscus is formed only on the perimeter of the plate. The plate does not have to be pulled above the surface to form the meniscus. The plate can rather be placed right at the surface of the liquid being measured, and not moved while surface tension is being measured.

These differences make the plate method vastly more accurate for determining the surface tensions of both pure liquids and surfactant solutions.

The lack of a pulled meniscus during the experiment means that the surface is does not have to be in flux (being stretched) during measurement. The plate is simply touched to the surface (typically dipped into the liquid and brought back and held within 1.0 microns of the surface position – in high-end tensiometers), and then the surface is allowed to relax and come equilibrium with the plate present – without being stretched or perturbed. Thus, if surfactants are present, they are given as long as they need to reach an equilibrated state – typically one minute is used, for most surfactant solutions, unless the surfactants are know to be exceedingly slow. Then the force on the plate is measured, and surface tension determined from the force. No non-equilibrium state of the surface is present during measurement, there are no corrections for volume of liquid hanging from the bottom of the plate (if the plate is flush with the surface) are necessary. Also, a Wilhelmy plate is easier to keep true and parallel with the surface – since it is a coupon of platinum, rather than a ring. Platinum is ductile enough to be straightened, and reformed, by hand, if necessary.

In the table below we add surface tension, measured by the plate method for the nonylphenol ethoxylate solution we have been discussing to the former ring method data presented.

Ring Pull-Through Rate During Surface Tension Measurement (mm/min)	Surface Tension Measured (corrected) Pure Water (mN/m)	Surface Tension Measured (corrected) Nonyl-Phenol Ethoxylate Solution (mN/m)
10	72.51	36.34
5	72.50	35.24
2	72.53	34.02
1	72.54	33.19
0.5	72.50	32.34
Zero – Plate Method	72.53	30.12

The 30.12 mN/m measurement of the surface tension of this solution can be verified by other methods (as discussed below) as the true equilibrium surface tension of this solution. The ring methods measurements are all higher, due to surface perturbation during the measurement of surface tension. And, this surfactant, in particular, nonylphenol ethoxylate is not one that would be considered to be particularly slow to adsorb at
surfaces. For other slower equilibrating surfactants, the increase in apparent surface tension measured by ring method would differ more widely.

Based on this, our laboratory never, unless specifically requested to do so to meet a customer’s standard or follow an ASTM standard – of which one exists calling for the ring method ATSM D971 – uses ring tensiometry as means of measuring equilibrium surface tension. The chance for error, particularly to the high side or surface tension determination, is just too great, unless you are working with a pure liquid, or a simple mixture, which you know will reach equilibrium surface tension in a short period of time.

Pendant Drop

Another method of measuring surface tension, considered even more accurate than plate method, is the pendant drop method. In this method, you use effectively no probe (ring or plate) for surface tension measurement. Surface tension is simply balanced against gravity. Gravity is your probe. So, the method is one step better than the plate method, in that it eliminates the possibly for a contaminated probe. It also offers the ability to controllably measure non-equilibrium surface tensions, which can be related to a particular surface age (unlike the non-equilibrium surface tensions which are measured by the ring method, which I consider just accidental and confusing).

The ability to measure non-equilibrium interfacial tension is important in high speed systems such as coatings and ink deposition, because the surface tension of the liquid at high speed (low surface age, after the surface of the liquid is created) is what determines how the liquid will first wet a substrate – not the surface tension of the liquid at equilibrium.

Perhaps this is has already become obvious, from our discussion of various surface tensions being measured on the same surfactant solution by the ring method, at variable rates. But, if not, here, we will make it clear. The surface tension of any dilute surfactant solution is not a single value. Surfactant solutions have an equilibrium surface tension, which is a single value representing how low the surface tension can get, if the surface is unperturbed for an indefinite period. And, that’s the surface tension that people talk about a surfactant solution having. However, in reality, the surface tension of a dilute surfactant solution is any (and every) value from the surface tension of pure water (72.5 mN/m) down to that equilibrium surface tension value – depending on how fast you create a new surface of the solution. If you create new surface so quickly that effectively no surfactant is at the surface – the surface tension will be 72.5 mN/m. Over time the surfactants go to the surface, and equilibrate there, and the surface tension decays to its equilibrium value. The rate of decay is important is a variety of industries. The pendant drop technique can measure surface tensions, at surface ages down to about 0.5 seconds. At lower surface ages a bubble pressure method is used. Let’s discuss pendant drop first.

The pendant drop technique works as follows. A drop of liquid to be studied for surface tension is formed on the end of a downward-pointing capillary tip. The drop is typically formed to about 90% of its detachment volume (from the capillary). The drop is then digitally imaged. The drop’s image is fit by a robust mathematical approach to determine the drop’s mean curvature at over 300 points along its surface.

The curvature of a drop that is pendant to a capillary tip, at any given point on its surface, is dependent on two opposing factors (or forces). Gravity works to make the drop elongated or “drip-like”. The greater the difference in density between the liquid and the outside gas (in this case air), the greater this force. Surface tension works to keep the drop spherical- since a sphere has the lowest surface to volume ratio of any shape, and surface tension is by definition the amount of work necessary to create a unit area of surface. Pendant drop surface tension evaluation involves observing the balance that exists between these two forces on a pendant drop, in the form of the drop’s mean curvature at various points along its surface. Lower surface tension means a more “drip-like” drop shape, higher surface tension means a more spherical drop shape.

The actual mathematics of pendant drop analysis are based on the Laplace equation which says that pressure differences exist across curved surfaces. The pressure difference at any given point on the surface (ΔP) is equal to mean curvature of the surface at that point ((1/r1 +1/r2), where r1 and r2 are the principal radii of curvature) multiplied by twice the tension (σ) contained in the surface.

ΔP = (1/r1 +1/r2) 2 σ

For a pendant drop, the pressure difference within the drop between any two vertical positions is:

Δρ g Z

where Δρ = the difference in density between the liquid that is forming the drop and the bulk gas, g = gravity, and Z = the vertical distance between the two positions, as shown below.

The measurement of surface tension is actually made by determining the mean curvature on the drop at over 300 points (like those labeled A and B above). Those points are then used in pairs, with the equations given above, to solve for surface tension. In the following manner:

( (1/r1 +1/r2)at A – (1/r1 +1/r2)at B ) 2 σ = Δρ g Zbetween A and B

Therefore, from one drop image, surface tension is determined at least 150 times. These surface tension values are averaged to give a single value for the overall surface tension of the drop. This technique has been found to be extremely accurate for determining surface tensions of liquids with known surface tension (typical errors of less than 0.1%). And, no probe is involved.

Of course, the disadvantage of the pendant drop method is that the densities of all liquids studied have to be predetermined to the same level of accuracy one expects for the surface tension data being measured. That, and the greater mathematical simplicity of the Wilhelmy plate method, make the pendant drop technique under appreciated, for most common work. However, it is probably, overall, the most accurate means of measuring true surface tension on any liquid – even high viscosity liquids – which have plate wetting problems in the Wilhelmy technique. The ability to run pendant drop experiments in a high temperature chamber also makes pendant drop the technique that our laboratory uses to measure surface tensions for molten polymers, hot melt adhesives, and even molten metals.

Pendant drop can also be either an equilibrium technique – just wait for the shape of the drop to stop changing after formation, and you can measure equilibrium surface tension. Or a non-equilibrium technique – form the drop quickly, and measure surface tensions while the shape of the drop is equilibrating. This is possible down to about 0.5 seconds of surface age, at moderate viscosity.

Below you will see graphical surface tension data, on the same surfactant solution we have been discussing, by three techniques. Pendant drop, Wilhelmy plate, and the bubble pressure method, which is the ultimate technique for low surface age surface tension.

Bubble Pressure Method

The principle of bubble pressure tensiometry is that gas (usually air or nitrogen) is blown through a small capillary tube which has one end submersed in the liquid to be tested and the other end attached to a pressure transducer. The gas forms bubbles in the liquid which initially grow on the end of the capillary, and eventually reach a volume at which they detach. During the formation of each bubble, a maximum gas pressure is reached. The maximum pressure is reached when the bubble is a hemisphere on the capillary, with a radius equal to the radius of the capillary tip. The maximum pressure is directly related to surface tension by the following equation:

σd = (Plax – Po) r
2

where:
σd = surface tension (dynamic)
Plax = maximum pressure
Po = hydrostatic pressure = (ρL – ρG) g d
ρL = density of liquid
ρG = density of gas
g = acceleration due to gravity
d = capillary immersion depth
r = capillary radius

Bubble pressure tensiometry is a non-equilibrium surface tension technique, rather than an equilibrium technique, like the Wilhelmy method, because the rate at which bubbles are formed can be made to vary. The faster bubbles are formed, the “younger” the gas/liquid surface is when its surface tension is measured. If surface active material(s) are present that take some time to diffuse to and come to equilibrium at a developing surface then the “older” a surface is, the lower its tension will be.

Bubble pressure data is typically either used as a means of studying the kinetics of diffusion and adsorption of surface active materials to surfaces, or used to mimic a dynamic application (examples: ink jet printing, spray coating, and roll coating). In the later case, people determine how fast liquid surfaces are created, in their processes, and what liquid surface tension is acceptable at that surface age. Then, they use bubble pressure tests to guide their formulating.

The graph below shows the spectrum of surface tensions that can be measured on a simple dilute surfactant solution, using multiple techniques. All of them are true tensions for the surface, at a given time after surface formation. Only the lowest one measurable are “equilibrium” surface tension values.

Conclusions

The graph above clearly shows that the nonyl-phenol ethoxylate solution, whose equilibrium surface tension is 30.12 mN/m has every surface tension from that of pure water to the equilibrium value, depending on the time frame for forming the surface. The pendant drop technique, is able to capture data from a surface tension of about 45 mN/m at 1.0 seconds of surface age, on down to equilibrium. The Wilhelmy technique is to be used only at equilibrium. And, the bubble pressure technique, as described below, captures surface tension back to about 5 milliseconds of surface age – critical if this were an ink, instead of just a surfactant solution, and we were doing high speed printing.

But the point, here, is to show that this simple surfactant solution does not reach an equilibrium surface tension for about 20 seconds (20,000 milliseconds), after a surface of it is formed. Other surfactants, in particular larger molecule surfactants, amphoterics, and fluorosurfactants are slower to reach that equilibrium value. Therefore, when seek to find the surface tension value for a surfactant solution, you have to ask what surface tension are you trying to measure? What is it you need to know about your surfactant solution?

If the surface is perturbed, and not at an equilibrium state during measurement, do you know that? And, is that what you mean to be measuring? In the case of ring method tensiometry, often, the answer is no. The three other methods all have better utility, depending on what information you seek.