Considerations Before Making Non- equilibrium Surface ... · profile of that drop isanalyzed for...

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Considerations Before Making Non-equilibrium Surface Tension Measurements

Technical Note #307eby:

James Chamberlain, KRÜSS USA,Dr. Christopher Rulison, Augustine Scientific

Abstract

There are many applications that can benefit from the measurement of non-equilibriumsurface tensions. These instances range from the basic study of surfactant diffusion andadsorption coefficients to the application of line-speed dependent coatings on movingsubstrates to the behavior of spray droplets in flight and upon adsorption to anappropriate substrate. The measurement of non-equilibrium/dynamic surface tensions bythe maximum bubble pressure method has been known for many years. However, themeasurement of non-equilibrium tensions by the Laplace solution to pendant drops hasonly recently become of interest due to the advent of higher speed computer processors.Three common methods for the measurement of surface tension (Wilhelmy plate method,pendant drop analysis, and the maximum bubble pressure method) are compared here forthe measurement of non-equilibrium tensions for various applications.

Liquid

Air

Plate madeof roughened Pt

F = Force, mN/m

L = WettedLength, mm

θ = contact angle= 0 for most liquids

against platinumθ

θ = 0

The Wilhelmy plate method is the mostprominent method for measuringequilibrium surface tensions of liquidsamples. A small platinum coupon isattached to a force-measuring sensor andprecisely dipped into a liquid andwithdrawn to the point where the bottomedge of the coupon is exactly parallel withthe surface of the bulk fluid itself. At thispoint, the surface tension is related to theforce exerted by the liquid wetting it bythe following equation:

σ = F / (cosθ * L)

where:

σ = surface tensionF = force

θ = contact angle of the sampleagainst the Pt coupon

L = wetted length of the Pt coupon

For almost all liquids against Pt, θ=0 andso the equation simplifies to:

σ = F / L

Although the plate method is mostcommonly used for measuringequilibrium surface tension, for certainsamples the plate method can be used tomeasure non-equilibrium surfacetensions. For examples, the timedependent surface tension lowering ofsolutions containing notoriously slowadsorbing florosurfactants can easily be

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measured by the plate method as shownbelow.

If we look closely at the time scale,however, we can see that this is certainlyan extremely slow adsorbing surfactantspecies. Neither solution has come to anequilibrium surface tension yet by thetime 10 minutes has passed. However,the majority of interesting time-dependent behavior can be witnessedwithin the first few minutes, which can beeasily measuring by the plate method.

Pendant Drop Analysis

The solution of the Laplace equation fordrops pending from a needle tip has alsobeen accomplished for a considerableperiod of time. For this method, a drop issimply pushed out of a syringe needleuntil it almost detaches from the needletip. At this point an image is snapped ofthe hanging drop and the contour of theprofile of that drop is analyzed for itsshape via the Laplace equation.

The actual mathematics of pendant dropanalysis are based on the fact thatpressure differences exist across curvedsurfaces. The pressure difference at anygiven point on the surface (∆P) is equal tomean curvature of the surface at thatpoint (1/r1 +1/r2), where r1 and r2 are theprincipal radii of curvature) multiplied bytwice the tension (σ) contained in thesurface.

∆P = (1/r1 +1/r2) 2 σ

For a pendant drop, the pressuredifference within the drop between anytwo vertical positions is:

∆ρ g Z

where ∆ρ = the difference in densitybetween the liquid that is forming thedrop and the bulk gas, g = gravity, and Z= the vertical distance between the twopositions, as shown below.

Those points are then used in pairs, withthe equations given above, to solve forsurface tension. In the following manner:

Historically the equation has been solvedby literally taking a photo of the drop andperforming the mathematics at a laterpoint by hand and calculator. Theequation is much more easily solved todaywith the advent of modern computers. Infact, the calculation is solved within theDSA 1 software for a few hundred pairsof points on a single drop in less than onesecond. This kind of speed allows one torapidly expel a new drop to the end of aneedle tip and follow the surface tension

z

A

B

( (1/r1 +1/r2)at A - (1/r1 +1/r2)at B ) 2 σ=

∆ρ g Zbetween A and B

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change (non-equilibrium surface tension)over relatively short time periods. Anexample of such data is given below.

Maximum bubble pressuretechnique:

The maximum bubble pressure techniqueis the most common technique formeasuring dynamic surface tensions.During the measurement, a gas (usuallynitrogen or air) is expelled through asmall orifice tube that is submersed intothe liquid sample of interest. The bubblesthat are blown out of the orifice have aback-pressure necessary to expel themfrom the tube and that back pressure isrelated to the surface tension by thefollowing form of the Laplace equation:

σd =0.5 (Plax - Po) r

where:

σd = surface tension (dynamic)r = capillary radiusPlax = maximum pressurePo = hydrostatic pressure (ρL - ρG) g dρL = density of liquidρG = density of gasg = acceleration due to gravityd = capillary immersion depthSo, for any given bubble the surfacetension of the solution of interest can bedetermined. The non-equilibrium aspectof the experiment is created by varyingthe rate at which bubble are formed atthe orifice. At the highest bubble ratesless time is allowed for surfactantaccumulation at the bubble surface and

thus shorter surface ages are realized. Atslow bubble formation rates, the amountof time for surfactant accumulation at thebubble surface is increased and theequation above then solves for surfacetension at longer surface ages. The datais usually plotted as surface tension vs.

log of surface age as shown below:

In practice, this surface age can be variedfrom about 5 ms to 1 minute. A fewphysical constraints of the measurementitself do not allow for surface agesoutside of these limits to be tested.

Surface Age Zero

For timed surface age measurements asshown in all of the above non-equilibriumexamples, the relative accuracy of aproper “time zero” for the starting pointof a newly formed, fresh surface isimportant to understand. The knownaccuracy of this time zero for eachmethod can also help provide a sense ofwhich experimental protocol is the mostappropriate.

For the plate method, the accurate timezero can be assumed to be the point atwhich the solution is first still. Normallythis would be after any pre-experimentalstirring. Although this situation iscomplicated by the small mixing thatoccurs when the plate is immersed intothe sample, we will assume that the zeropoint in time occurs when mixing isstopped. Depending on the procedureused, the time between the stoppage of

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any mixing and the realization of the firstmeasurement can range between 5-30seconds. So, for non-equilibriummeasurements in the time range ofminutes, this delay and inaccuracy of timezero is acceptable. For shorter surfaceages, the delay in the measurement itselfoutweighs the length of themeasurement making the plate methodnot-useful for measuring shorter agednon-equilibrium tensions.

The time necessary to produce a new,fresh pendant drop for subsequentLaplace solution of non-equilibriumsurface tensions is approximately 0.2-2seconds depending on the method usedfor producing the drop. This shorter timeframe and accuracy of a point of timezero allows for more accuracy in themeasurement range of seconds to aminute. However, the delay in surfaceproduction still does not allow for surfaceages under a few seconds to beaccurately determined.

The bubble pressure method intrinsicallyaccurately determines a time zero for anew, fresh surface. Time zero isdetermined to be the time at which theproceeding bubble released from theorifice. The time between this point andthe point of surface tension measurementis accurately determined in the regime ofmicroseconds. This accurate time zeroallows for accurate surface tension curvesdown to surface ages as low as 5 ms.

Which Method Should YouChoose?

All 3 methods described above arecompared for a very common PhenolEthoxylate surfactant in the curve below:

As the curve demonstrates, the 3methods show nicely contiguous data. Infact, the pendant drop data takes overwhere the bubble pressure technique

reaches it’s upper limit in surface age.The Wilhelmy plate data continues on atlonger time frames shown by the pendantdrop method. In essence, the plotanswers the question of which method tochoose on its own. Samples that requirequick data acquisition and a well definedtime zero must be run by the bubblepressure method. Samples that do notrequire such speed can be run on and, insome cases, must be run by Pendant dropanalysis. Very slow surface ages andsamples that do not change surfacetension within a minute or so can still berun by the Wilhelmy plate method, butthere are few of these such samples.

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Summary

Choose your technique andinstrumentation carefully when pursuingnon-equilibrium surface tension data.Below is a short key for generalconsiderations:

With these basic rules in mind and afundamental understanding of themeasurements, you can make a moreinformed decision toward the mostappropriate instrumentation, giving you

the clearest path to solving your surfacescience problem!

Bubble Pressure Pendant Drop Wilhelmy Plate

basic surfactant kinetics

ink jet studies

sprays

fast application of coatings

viscous inks

slow adsorbing surfactants

slow application of coating

ONLY for very slow adsorbingsurfactant systems

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