Direct measurement of the attenuation of capillary waves by laser interferometry:Noncontact determination of viscosityF. Behroozi, B. Lambert, and B. Buhrow Citation: Applied Physics Letters 78, 2399 (2001); doi: 10.1063/1.1365413 View online: http://dx.doi.org/10.1063/1.1365413 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/78/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stokes’ dream: Measurement of fluid viscosity from the attenuation of capillary waves Am. J. Phys. 78, 1165 (2010); 10.1119/1.3467887 Using laser Doppler vibrometry to measure capillary surface waves on fluid-fluid interfaces Biomicrofluidics 4, 026501 (2010); 10.1063/1.3353329 Measurement of the dispersion relation of capillary waves by laser diffraction Am. J. Phys. 75, 896 (2007); 10.1119/1.2750379 Direct measurement of the dispersion relation of capillary waves by laser interferometry Am. J. Phys. 74, 957 (2006); 10.1119/1.2215617 Surface tension measurement technique by differential phase detection of capillary waves in liquids Rev. Sci. Instrum. 71, 4231 (2000); 10.1063/1.1315349

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Direct measurement of the attenuation of capillary waves by laserinterferometry: Noncontact determination of viscosity

F. Behroozi,a) B. Lambert, and B. BuhrowDepartment of Physics, University of Northern Iowa, Cedar Falls, Iowa 50614

~Received 22 September 2000; accepted for publication 19 February 2001!

The determination of viscosity from the damping of capillary waves has been of great interest, as itaffords the possibility of measuring viscosity without contact with the fluid. Here we describe anoncontact method for precision measurement of the amplitude of capillary waves on fluids. Thetechnique utilizes a miniature laser interferometer to map the wave profile with a resolution of about10 nm. We use this technique to obtain the dispersion and attenuation of capillary waves on wateras a test case. Furthermore, the attenuation data is used to obtain the viscosity of water as a functionof temperature. ©2001 American Institute of Physics.@DOI: 10.1063/1.1365413#

Surface waves on fluids, with wavelengths in the milli-meter range, are known as capillary waves. In this waveregime surface tension and viscosity govern the propagationand attenuation of surface waves while gravity plays a minorrole. Therefore, data on dispersion and attenuation of capil-lary waves may be used to determine the surface tension andviscosity of fluids.1–8 Of particular interest has been the de-termination of viscosity from the damping of surfacewaves.9–13 Here we describe a noncontact method for preci-sion measurement of the profile and attenuation of capillarywaves on fluids.14–16The technique utilizes a miniature laserinterferometer17 to obtain the wave profile with a resolutionof about 10 nm—some 50 times better than the resolution ofa typical optical microscope. We use this technique to obtainthe dispersion and attenuation of capillary waves on purewater as a test case. Furthermore, the attenuation data is usedto obtain the viscosity of pure water as a function of tem-perature. The results are in agreement with the acceptedvalues18 obtained by traditional flow viscometry, thus dem-onstrating the utility of this method for measuring viscosity.

The capillary waves are generated electronically by plac-ing a metallic blade a few tenths of a millimeter above thewater surface. A dc-biased sinusoidal voltage of about a hun-dred volts at a selected frequency is applied between theblade and the water. Since water molecules are polar, thealternating electric field under the blade generates two cap-illary wave trains that recede from the blade on the two sides.Typically the amplitude of these waves is of the order of onemicron.

A standing wave is generated when two blades, sepa-rated by a few centimeters, are used to generate waves of thesame phase, amplitude, and frequency. Since each bladesends a wave train toward the other, a standing capillarywave is established on the water surface between the twoblades. If the distance between the two blades is chosen to bea half odd-integer wavelength, the two wave trains interferedestructively on the outer sides of the blades. This judiciouschoice of the blades’ separation produces a region of stand-ing waves between the blades while the surface outside theblades remains calm.

To obtain the wave profile we use a fiber-optic detectionsystem that functions as a miniature laser interferometer. Theheart of the system consists of a single mode optical fiber,one end of which is positioned a short distance above thefluid surface~Fig. 1!. Laser light, traveling through the opti-cal fiber is partially reflected from the cleaved tip of the fiberand again from the fluid surface. The two reflected beamstravel back through the same fiber forming an interferencepattern. As the water level changes due to the wave motion,the interference signal portrays an accurate record, in realtime, of the variation of the gap.

The interference signal is detected, amplified, and digi-tized for analysis. In Fig. 2, at the top of the frame, we showa typical raw interference pattern taken directly from the de-tector as displaced on a digital oscilloscope. The solid graphis a mathematical fit to the scope trace, and the dashed curveis the surface wave that produced the interference pattern.Since there is a one-to-one correspondence between the sur-face wave and the resulting interference pattern, the profileof the surface wave can be recovered by an analysis of theinterference pattern. Indeed, as discussed shortly, the numberof fringes in the interference pattern is directly proportionalto the amplitude of the surface wave. For the pattern shown

a!Electronic mail: behroozi@uni.edu

FIG. 1. Schematic of the fiber probe above a capillary wave. The fiber-opticprobe is mounted on an electronic micrometer, which records the probeposition on the surface with an accuracy of 1mm. The distance from the tipof the probe to the equilibrium surface isd0 , and the roundtrip distancebetween the tip of the probe and the wave surface, i.e., the path differencebetween the two reflected beams, isD. A typical wavelength is about 1 mm,and typical wave amplitudes are less than 1mm. The wave amplitude hasbeen vastly exaggerated for clarity.

APPLIED PHYSICS LETTERS VOLUME 78, NUMBER 16 16 APRIL 2001

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in Fig. 2, the number of fringes is 6.78 and the amplitude ofthe wave is 1.07 micron.

To determine the wave amplitude of a traveling wavefrom the interference record, the vertical oscillation of thewater surface under the probe may be represented by

y~ t !5a sin~vt1b!. ~1!

Herea is the wave amplitude,v is the angular frequency,t isthe time, andb is a phase which depends only on the posi-tion of the probe relative to the blade. Ifd0 is the air gapbetween the probe and the equilibrium surface of the water,then the path difference between the two reflected beams is

D52@d02a sin~vt1b!#. ~2!

Thus, the ac component of the resulting interference patternis given by

Y~ t !5A cos@~2pD/l l !1p#, ~3!

whereA is the amplitude of the interference signal,l l is thewavelength of the laser light, andp is added to the phase toaccount for the fact that the light beam reflecting from thewater surface suffers a phase shift ofp radians. When Eq.~2! is substituted in Eq.~3!, the result is

Y~ t !5A cos$~4p/l l !@d02a sin~vt1b!#1p%, ~4!

which simplifies to

Y~ t !5A cos@b sin~vt1b!2w#. ~5!

Hereb stands for 4pa/l l , andw5(p14pd0 /l l).The relationship betweena andb determines the relation

between the wave amplitude and the number of fringes in theinterference signal. Thus,

a5bl l /4p. ~6!

To determine the amplitude of the capillary wave from theinterference data, it is only necessary to extract the parameter

b from the data to use in Eq.~6!. This is achieved by fittingthe analytical expression in Eq.~5! to the interference data.To accomplish this, the interference pattern is digitized andused as input in a multi-variable fit routine, which adjusts thefour parameters of Eq.~5! until a beneficial fit is achieved.Indeed, of the four parameters in Eq.~5!, A andb are readilyavailable from the raw data, so the fit routine reduces to asearch in the two-parameter space ofb and w. By thismethod we determine the attenuation of the wave amplitudeas a function of the distance traveled from the source. Buthow is the wave attenuation related to viscosity?

The dissipation of wave energy due to viscosity mani-fests itself in the attenuation of the amplitude as the wavetravels along the surface. Indeed, in the linear limit the rateof energy loss is proportional to the wave energy19 and isgiven by

dE/dt52~4k2h/r!E, ~7!

whereE is the wave energy per unit surface area,h is thefluid viscosity, r is the fluid density, andk52p/l is thewave number. Since the wave energy is proportional to thesquare of its amplitude, we can immediately write Eq.~7! interms of the wave amplitude,

da/a52~2k2h/r!dt. ~8!

Equation~8! implies that the fractional loss of the wave am-plitude is proportional to the elapsed time. Since in a timeintervaldt the wave train travels a distancedx5vgdt, wherevg is the group velocity, Eq.~8! can be recast into

da/a52~2k2h/r!dx/vg . ~9!

This immediately yields,

a5a0e2ax, ~10!

wherea, the attenuation coefficient, is given by

a5~2k2h/rvg!. ~11!

For capillary waves on deep water the group velocityvg

is given by20

vg5~g13sk2/r!/2~gk1sk3/r!1/2. ~12!

When Eq.~12! is used in Eq.~11!, we obtain the viscosity interms of four measurable quantities, namely, wave numberk52p/l, surface tensions, densityr, and the attenuationcoefficienta. Indeed, we have

h5@ar/2k2#@~g13sk2/r!/2~gk1sk3/r!1/2#. ~13!

The attenuation coefficienta is obtained from a plot of thewave amplitude versus the distance from the blade. Thewavelengthl and the surface tension are determined fromthe dispersion data as described.

To obtain the dispersion data, a standing wave is estab-lished on the fluid surface. Measurement of the distance be-tween several nodes yields the wavelength of the capillarywave. In our setup, the fiber optic probe is attached to amicro-positioner, which in turn is equipped with a digitalmicrometer. This enables us to measure the wavelength ofthe standing capillary waves routinely to within a micron. Aprecise dispersion relation is obtained by measuring thewavelength at various frequencies.

FIG. 2. The raw interference signal for a half period is shown at the top. Thesolid curve is the mathematical fit@Eq. ~5!#, which replicates the trace faith-fully. The dashed curve gives the vertical displacement of the water waveunder the probe as a function of time. The right scale is for the interferencesignal and the left scale is for the water wave. In the case shown the waveamplitude is 1.07mm and the wave frequency is 160 Hz.

2400 Appl. Phys. Lett., Vol. 78, No. 16, 16 April 2001 Behroozi, Lambert, and Buhrow

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Figure 3 shows the phase velocity of capillary waves onwater at 20 °C as a function of wavelength obtained by thismethod. The solid line is a plot of the simplified dispersionrelation of surface waves on water20 given by

vw5~gl/2p12ps/lr!1/2, ~14!

wherevw is the phase velocity,g is the acceleration of grav-ity, l is the wavelength,s is the surface tension, andr is thefluid density. We have used the widely accepted values ofthe parameterss ~72.8 dyn/cm! and r ~0.9982 gm/cm3! forpure water at 20 °C to plot the solid line in Fig. 3. Con-versely, the dispersion data can be fitted to Eq.~14! to extracts.

Figure 4 gives the wave amplitude versus distance fromthe source. To obtain this data two probes are utilized, onestationary, the other moving. The amplitude data obtained bythe moving probe is a function of position and is normalizedby that obtained from the stationary probe to account for anychange in the equilibrium water level due to evaporation.While a small change in the water level has no effect on thedetection system, it does affect the amplitude of the water

waves being generated under the blade. The normalizationprocedure outlined before eliminates this source of error. Thesolid graph in Fig. 4 is an exponential fit to the data andgives the attenuation coefficienta for use in Eq.~13!.

Figure 5 gives our measured values of the kinematicviscosity, h/r, versus the temperature for pure water. Thesolid line in the figure is a second order polynomial fit to thepublished data8 for pure water and is included for compari-son. Since water has a very small viscosity to begin with,measuring the temperature variation of its viscosity consti-tutes a severe test of our method. The results presented inFig. 5 show that the noncontact method described here pro-vides a sensitive new alternative to flow viscometry. Further-more, the noncontact nature of the method provides anotherclear advantage by eliminating the possibility of contamina-tion of the fluid under study.

Financial support from the Carver Trust, UNI AppliedTechnology Fund, and the Iowa Space Grant Consortium aregratefully acknowledged.

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FIG. 3. Dispersion of capillary waves on water at 20 °C. The solid line isthe simple theory, which ignores the viscosity. The viscosity of water is sosmall that its influence on the dispersion relation is not evident at this scale.

FIG. 4. A plot of wave amplitude vs distance from the source. The solid lineis an exponential fit to the data. This plot yields a value of 8.931023 cm2/s for the kinematic viscosity of water, in agreement with theaccepted value obtained with conventional flow viscometry.

FIG. 5. Kinematic viscosity vs temperature for pure water. The data pointsare shown as open circles and the solid line is a fit to the published data.

2401Appl. Phys. Lett., Vol. 78, No. 16, 16 April 2001 Behroozi, Lambert, and Buhrow

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