## Recipe for Dilational Stress Measurements

### Definitions

Dilational stress measurements are made by simultaneously analyzing the interfacial tension and surface area of a drop whose volume is periodically perturbed. The perturbation (changes) must be regular and repeated; that is what we mean by periodic. When this is true, we can use Fourier analysis to extract the dilational stress.

These measurements are much like other materials science measurements: we apply a stress, a force, and measure the strain, a change in shape or length. More generally, we apply a "dose" and measure a "response." The result is a complex number (a vector) called the modulus. For a more complete discussion, see

**Dynamic Surface Tension and Dilational Stress Measurements
Using the Drop Shape Method**

in the Papers section of the FTA website. A very practical, interesting, and easily available sample for dilational stress work is milk (obtained from a store near you). An application note on milk and the more general topics of noise and resolution in dilational stress measurements

**Milk as an Example of Dynamic Interfacial Tension and
Dilational Stress**

is also in the Papers section.

The mathematics is based on a sinusoidal (sine-wave) variation of dose, which in this case is the surface area of the drop. However the Fourier transforms we use have the advantage of being excellent filters so we can effectively ignore the non-sinusoidal components of dose and response.

### Movie Setup

The first task is to determine what range of drop volumes are suitable. Then we will program the pump to repetitively make these volumes.

Notice we intend to perturb drop volume and then assume the resulting drop shape perturbation is satisfactory. Even if we cause a sinusoidal drop volume, the drop shape will not be exactly sinusoidal. Volume varies as millimeters cubed and area as millimeters squared, so they do not track linearly. This is not a problem, however, using the Fourier transform filtering.

The practical way to make the measurement is with a pendant drop. The sessile drop method depends too much on the surface upon which the drop rests. You must

Empirically determine the largest drop volume that will reliably hang on the dispense needle. You may wish to use a larger needle than normal. Watch the effects of vibration at this volume. As an example, assume this volume is 12ul.

Empirically determine the smallest drop volume that will yield a reliable interfacial tension measurement. Recall that the drop must be large enough to be distorted by gravity. As an example, assume this volume is 7ul.

Determine how fast you wish to perturb the drop volume. This may require some trial-and-error. In fact you may need to perform a complete experiment to find the answer. As a place to start, consider several seconds per cycle. As an example, let us choose 10s. This will be 5s to expand the drop and 5s to retract the drop to make one cycle. What really matters is the relative rate at which the surface area is expanded or contracted. Thus you achieve the same "dose" by expanding over a large percentage change at a slow rate as you do a small change at a fast rate.

You now have a pump rate: you have the change in volume from largest to smallest and you have the time to do it. In our example, the change in volume is 5ul (12ul - 7ul) and the time for each phase is 5s. Therefore the pump rate is 1ul/s (5ul/5s). Pump rates must be slow, in this range, in order not to introduce significant vibration. Experience will show you what you can tolerate.

Setup the Pump Program so that initially it dispenses to the maximum possible volume, 12ul in this case, then starts its cycle: 5ul in (negative volume with time = 5s) and then 5ul in 5s back out. This will be one "loop." What you have programmed is a "triangle wave" instead of a sine wave, but it is close enough for practical purposes. You can make better approximations, if you find them useful, after you have the other parameters affirmed.

The advantage of initially pumping to the maximum volume is that it assures you that the drop will remain attached and not fall off. It also offers a second alternative: You can manually pump to the maximum volume and then have the pump program simply perform perturbations of subtracting and then returning volume to the drop. This lets you manually establish the beginning drop without having to change the pump program. It is a matter of determining which is easiest for you.

Decide how many loops to execute. This will set the overall time for the experiment. Use no less than 4 loops. You may find a number like 15 or 25 more useful. Enter this in the pump program.

When the pump program runs, it will measure the actual time to execute each loop. This is useful because the pump is digital and can only execute specific rates. These may be close to but not exactly equal to what you request. Secondly, there is some time overhead when you change the pump speed or direction. These matters become evident at overall loop times in the range of a second or less. Slower loops will be more accurate. When you perform the dilational stress analysis, it will be useful to know the perturbation frequency or period. You can get the actual period from this display.

Finally, you must setup movie capture. You will want a minimum of 4 images per loop, with 8 to 20 better. Take the programmed loop time, 10s in our example, and divide by the selected number of images to get the image period. It does not matter that the actual loop time will be slightly different. The Fourier transform will sort this out. You will not use Images before Trigger (leave at the default 1, with Image Period before Trigger = .033s). The images you acquire will all be after the trigger. You can manually click Trigger after you start the Run or you can use the Video Trigger and set the cross-hairs target so it will be hit when the initial drop gets to its maximum. Set the pump for Start on Run but not Stop on Trigger.

Acquire the movie.

For additional information on pumping and the resultant volume and surface area waveforms, see

**Dilational Stress Pump Waveforms**

and

For information on computer hardware requirements in order to acquire data in real time, see

**Computer Speed and Dilational Stress Analysis**

### Movie Analysis

Analyze the movie for interfacial tension. Check the graph and observe that the drop surface area curve shows the regular perturbation you expected. It is OK if the surface tension curve is "noisy," but the surface area curve should be relatively clean.

It is also possible to acquire the data using Live analysis and Data Log. You File | Import the Data Log, which converts it to a movie with data but no images. The advantage is that the files are small. On a practical basis, however, you can only do this after you have at least one normal movie to validate your parameter selections.

### Dilational Stress Analysis

With an analyzed movie in hand, move to the Dynamic IFT | Dilational Stress tab. You will have several choices to make:

Calculate over all data or only over a limited range specified by image indexes. Inspect the movie Graph to see whether the data appears "clean" over the entire range. If so, use the entire range. But consider the following possibility. Say the drop falls off part way through the experiment. Then you could use the first part but not the last part.

Subdivide the data. Always start with no subdivisions and see what the dilational stress calculations yields. If the Graph data appears smooth, without too much noise, you may consider subdivisions. This means chopping the existing movie into several smaller ones for calculation purposes. This is useful if you think the data may be slowly changing. This sometimes called "ageing." The number of subintervals should never be so large that the number of loops or cycles per subdivision falls below 4.

Click Calc to perform the calculation, which may take a few seconds.

The grid in the upper right part of the tab will show the results for each subdivision. The Fourier transforms for the surface tension and surface area functions are shown below. If there is more than one interval, click in that column to see its Fourier transform graphs.

The Fourier transform graphs show the "quality" of the data. The data you want is at the perturbation frequency of the original pumping cycle. The question is how much it stands above data at other frequencies, which may be considered noise. The numeric data in the grid is calculated from the peak at the perturbation frequency. If the calculated perturbation frequency, or period, is much different from the loop time, it is likely noise has overwhelmed the signal and the experiment is not useful.

As a final word of advice, always start with moderate or slow perturbation rates. As you increase the rate, vibration will increase, introducing noise, and the interfacial tension "response" may fall so far behind one cycle that it appears in the following cycle, which confuses interpretation.