Abstract

A new generation of vibration-mitigating surface-light-scattering instrumentation has been designed and built. The computational application of an instrument function derived by use of Fourier optics is presented. This instrument and its accompanying suite of analysis software allow us to easily make accurate and noninvasive measurements of the interfacial tension, volume viscosity, and other interfacial parameters of fluids. We derived the necessary surface response function algorithms to study both simple fluids and binary fluids at their wetting transition and near their critical points. These developments can be applied to study systems with liquid–vapor and liquid–liquid interfaces, including spread monolayers, whenever optical access for a laser beam is available.

© 2001 Optical Society of America

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References

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  1. R. V. Edwards, R. S. Sirohi, J. A. Mann, L. B. Shih, L. Lading, “Surface fluctuation scattering using grating heterodyne spectroscopy,” Appl. Opt. 21, 3555–3568 (1982).
    [CrossRef] [PubMed]
  2. L. Lading, J. A. Mann, R. V. Edwards, “Analysis of a surface-scattering spectrometer,” J. Opt. Soc. Am. A 6, 1692–1701 (1989).
    [CrossRef]
  3. W. V. Meyer, J. A. Lock, H. M. Cheung, T. W. Taylor, P. Tin, J. A. Mann, “A hybrid reflection–transmission surface light scattering instrument with reduced sensitivity to surface sloshing,” Appl. Opt. 36, 7605–7614 (1997).
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill Classic Textbook Reissue Series (McGraw-Hill, New York, 1968), p. 287.
  5. J. A. Mann, “Surface light scattering spectroscopy,” in Proceedings of the NATO Advanced Research Workshop on Light Scattering and Photon Correlation Spectroscopy, Vol. 40 of NATO ASI Series, 3. High Technology, E. R. Pike, J. B. Abbiss, eds. (Kluwer, Dordrecht, 1997), p. 97–115.
  6. D. Fenistein, G. H. Wegdam, W. V. Meyer, J. A. Mann, “Capillary waves on an asymmetric liquid film of pentane on water,” Appl. Opt. 40, 4134–4139 (2001).
  7. J. A. Mann, P. D. Crouser, W. V. Meyer, “Surface fluctuation spectroscopy by surface-light-scattering spectroscopy,” Appl. Opt. 40, 4092–4112 (2001).
    [CrossRef]
  8. D. Langevin, ed., Light Scattering by Liquid Surfaces and Complementary Techniques, 1st ed., Vol. 41 of Surfactant Science Series, M. J. Schick, F. M. Fowkes, eds. (Marcel Dekker, New York, 1992).
  9. A. E. Smart, R. V. Edwards, W. V. Meyer, “Quantitative simulation of errors in correlation analysis,” Appl. Opt. 40, 4064–4078 (2001).
    [CrossRef]
  10. J. A. Mann, “Dynamics, structure and function of interfacial regions,” Langmuir 1, 10–23 (1985).
    [CrossRef]
  11. R. S. Hansen, J. A. Mann, “Propagation characteristics of capillary waves,” J. Appl. Phys. 35, 152–158 (1964).
    [CrossRef]
  12. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999). Note that formula 4.5.9 for determining the weights ωin is missing a factor of 2 in the numerator. This can be confirmed by comparing ∫-∞+∞ exp[-x2]x2 dx = π1/2/2 to ∑i=-nn ωinf(uin); other even powers of u in f(x) = xu will also show this. Moreover, π1/2 = ∑i=-nn ωin to 13 significant figures.
  13. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  14. C. L. Yaws, “Library of physico-chemical-property data,” in Handbook of Viscosity (Gulf, Houston, Tex., 1997). Note that other literature values for acetone indicate that the correction shown below is needed for the exponent for the Yaws acetone surface tension extrapolation formula:surfaceTensionLitValue=surfaceTensionAtRefTempInKelvin×critcalTempInKelvin − tempInKelvincritcalTempInKelvin − refTempInKelvin1.4728.
  15. R. V. Edwards, Western, Case Western Reserve University, Cleveland, Ohio (personal communication, 1985).
  16. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

2001 (3)

1997 (1)

1989 (1)

1985 (1)

J. A. Mann, “Dynamics, structure and function of interfacial regions,” Langmuir 1, 10–23 (1985).
[CrossRef]

1982 (1)

1964 (1)

R. S. Hansen, J. A. Mann, “Propagation characteristics of capillary waves,” J. Appl. Phys. 35, 152–158 (1964).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Cheung, H. M.

Crouser, P. D.

Edwards, R. V.

Fenistein, D.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999). Note that formula 4.5.9 for determining the weights ωin is missing a factor of 2 in the numerator. This can be confirmed by comparing ∫-∞+∞ exp[-x2]x2 dx = π1/2/2 to ∑i=-nn ωinf(uin); other even powers of u in f(x) = xu will also show this. Moreover, π1/2 = ∑i=-nn ωin to 13 significant figures.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill Classic Textbook Reissue Series (McGraw-Hill, New York, 1968), p. 287.

Hansen, R. S.

R. S. Hansen, J. A. Mann, “Propagation characteristics of capillary waves,” J. Appl. Phys. 35, 152–158 (1964).
[CrossRef]

Lading, L.

Lock, J. A.

Mann, J. A.

Meyer, W. V.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999). Note that formula 4.5.9 for determining the weights ωin is missing a factor of 2 in the numerator. This can be confirmed by comparing ∫-∞+∞ exp[-x2]x2 dx = π1/2/2 to ∑i=-nn ωinf(uin); other even powers of u in f(x) = xu will also show this. Moreover, π1/2 = ∑i=-nn ωin to 13 significant figures.

Shih, L. B.

Sirohi, R. S.

Smart, A. E.

Taylor, T. W.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999). Note that formula 4.5.9 for determining the weights ωin is missing a factor of 2 in the numerator. This can be confirmed by comparing ∫-∞+∞ exp[-x2]x2 dx = π1/2/2 to ∑i=-nn ωinf(uin); other even powers of u in f(x) = xu will also show this. Moreover, π1/2 = ∑i=-nn ωin to 13 significant figures.

Tin, P.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999). Note that formula 4.5.9 for determining the weights ωin is missing a factor of 2 in the numerator. This can be confirmed by comparing ∫-∞+∞ exp[-x2]x2 dx = π1/2/2 to ∑i=-nn ωinf(uin); other even powers of u in f(x) = xu will also show this. Moreover, π1/2 = ∑i=-nn ωin to 13 significant figures.

Wegdam, G. H.

Western,

R. V. Edwards, Western, Case Western Reserve University, Cleveland, Ohio (personal communication, 1985).

Yaws, C. L.

C. L. Yaws, “Library of physico-chemical-property data,” in Handbook of Viscosity (Gulf, Houston, Tex., 1997). Note that other literature values for acetone indicate that the correction shown below is needed for the exponent for the Yaws acetone surface tension extrapolation formula:surfaceTensionLitValue=surfaceTensionAtRefTempInKelvin×critcalTempInKelvin − tempInKelvincritcalTempInKelvin − refTempInKelvin1.4728.

Appl. Opt. (5)

J. Appl. Phys. (1)

R. S. Hansen, J. A. Mann, “Propagation characteristics of capillary waves,” J. Appl. Phys. 35, 152–158 (1964).
[CrossRef]

J. Opt. Soc. Am. A (1)

Langmuir (1)

J. A. Mann, “Dynamics, structure and function of interfacial regions,” Langmuir 1, 10–23 (1985).
[CrossRef]

Other (8)

D. Langevin, ed., Light Scattering by Liquid Surfaces and Complementary Techniques, 1st ed., Vol. 41 of Surfactant Science Series, M. J. Schick, F. M. Fowkes, eds. (Marcel Dekker, New York, 1992).

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill Classic Textbook Reissue Series (McGraw-Hill, New York, 1968), p. 287.

J. A. Mann, “Surface light scattering spectroscopy,” in Proceedings of the NATO Advanced Research Workshop on Light Scattering and Photon Correlation Spectroscopy, Vol. 40 of NATO ASI Series, 3. High Technology, E. R. Pike, J. B. Abbiss, eds. (Kluwer, Dordrecht, 1997), p. 97–115.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999). Note that formula 4.5.9 for determining the weights ωin is missing a factor of 2 in the numerator. This can be confirmed by comparing ∫-∞+∞ exp[-x2]x2 dx = π1/2/2 to ∑i=-nn ωinf(uin); other even powers of u in f(x) = xu will also show this. Moreover, π1/2 = ∑i=-nn ωin to 13 significant figures.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

C. L. Yaws, “Library of physico-chemical-property data,” in Handbook of Viscosity (Gulf, Houston, Tex., 1997). Note that other literature values for acetone indicate that the correction shown below is needed for the exponent for the Yaws acetone surface tension extrapolation formula:surfaceTensionLitValue=surfaceTensionAtRefTempInKelvin×critcalTempInKelvin − tempInKelvincritcalTempInKelvin − refTempInKelvin1.4728.

R. V. Edwards, Western, Case Western Reserve University, Cleveland, Ohio (personal communication, 1985).

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

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Figures (16)

Fig. 1
Fig. 1

Overcoming surface sloshing. Comparison of the differences in beam spot stability at the detector for a reflected laser beam propagating from the fluid interface of (a) a flat dish and (b) a cylindrical cell. Note the grating orientation in (b).

Fig. 2
Fig. 2

Drawings to aid in the visualization of how cross correlation is used to suppress the affects of a moving laser beam with a Gaussian beam profile. Beam movement would introduce an intensity fluctuation at the detector.

Fig. 3
Fig. 3

The surface-light-scattering instrument shown uses a surface reflection mode. In this layout the grating is imaged onto the fluid–fluid interface by a 50-mm lens placed 1 focal length from both the grating and slit, followed by a 150-mm lens placed 1 focal length from the slit and 1 focal length (accounting for the index of refraction) from the fluid–fluid interface. This results in a grating spot-size magnification of 3, perpendicular to the lines of the phase grating.

Fig. 4
Fig. 4

Surface-light-scattering instrument layout using TIR.

Fig. 5
Fig. 5

Correlograms for water, T = 20 °C, q = 500 cm-1, with different N values illustrate the effect of the instrument function. Most spectrometers operate between N = 20 and N = 30. For some experiments N down to ∼10 allows the spot size imaged on the surface to be smaller than 1 mm.

Fig. 6
Fig. 6

Fit of the normal probability distribution function (PDF) to the experimental PDF estimated from replicate data of the correlogram for toluene. The data are shifted slightly from zero at PDF = 0.5.

Fig. 7
Fig. 7

Experimental PDF computed from replications of the correlogram for toluene. Three PDF estimations were made: The residuals for the first three maxima beyond R i (0), the first three minima, and three lag-times where R i was close to zero.

Fig. 8
Fig. 8

(a) The Edwards weight factor Λ n = 1/σn2 used in the maximum likelihood fitting algorithm is compared with the correlogram for water. (b) The channel variance computed from multiple runs of correlograms is compared with 1/Λ n as computed in a least-squares fitting program first written by Edwards. The two curves were displaced slightly for this comparison. A small systematic variation in the baseline was subtracted from the residuals before plotting.

Fig. 9
Fig. 9

Acetone data taken with cross correlation. Data fit with an exponentially damped cosine model and no instrument function: ω = 9756.85 Hz; Γ = 488.723 Hz; surface tension, 23.244 dynes/cm (mN/m); viscosity, 0.53082 cP; data, ■; fit, ●; residuals, ao-40-24-4113-i001.

Fig. 10
Fig. 10

Acetone data taken with cross correlation. Data fit with an exponentially damped cosine model and the Mann GHG (MGHQ) instrument function: ω = 9780.51 Hz; Γ = 304.265 Hz; surface tension, 23.194 dynes/cm (mN/m); viscosity, 0.31703 cP; data, ■; fit, ●; residuals, ao-40-24-4113-i002.

Fig. 11
Fig. 11

Acetone data taken with cross correlation. Data fit with a Lorentzian model and no instrument function: ω = 9756.70 Hz; Γ = 488.983 Hz; surface tension, 23.244 dynes/cm (mN/m); viscosity, 0.53113 cP; data, ■; fit, ●; residuals, ao-40-24-4113-i003.

Fig. 12
Fig. 12

Acetone data taken with cross correlation. Data fit with a Lorentzian model and the MGHQ instrument function: ω = 9780.18 Hz; Γ = 305.733 Hz; surface tension, 23.194 dynes/cm (mN/m); viscosity, 0.31869 cP; data, ■; fit, ●; residuals, ao-40-24-4113-i004.

Fig. 13
Fig. 13

Acetone data taken with autocorrelation. Data fit with a Lorentzian model and the MGHQ instrument function: ω = 9753.07 Hz; Γ = 347.622 Hz; surface tension, = 23.098 dynes/cm (mN/m); viscosity, 0.36639 cP. Notice the low-frequency band of noise to the left of the peak when not cross correlating as in Fig. 12–and this is a fairly clean data set. Data, ■; fit, ●; residuals, ao-40-24-4113-i005.

Fig. 14
Fig. 14

Acetone data taken with cross correlation. Data fit with both the Mann and Meunier SRFs along with the MGHQ instrument function. Identical results were obtained: surface tension, 23.270 dynes/cm (mN/m); viscosity, 0.31853 cP; data, ■; fit, ●; residuals, ao-40-24-4113-i006.

Fig. 15
Fig. 15

Power spectra measurements of a thin film of pentane on water, where the film thickness changes with temperature. These data were taken with TIR (see Fig. 4) and plotted parametrically as functions of frequency and temperature, with the temperature axis being plotted nonlinearly. Compare with the theory plot shown in Fig. 16, noting that Fig. 16 does not include instrument function broadening. Gaps in the experimentally measured power spectra are visible where one expects a large low-frequency contribution, e.g., at 45 °C. Cross correlation was used along with roll-off filters set for -6 dB at 100 Hz and 1 MHz.

Fig. 16
Fig. 16

Theoretical power spectra for a thin film of pentane on water. These power spectra are calculated without free parameters and are given as a function of interfacial thickness for a fixed computational temperature of 45 °C, a Hamaker coefficient of 1.1 × 10-21 J, and K = 1000 cm-1. Roll-off filters were not used in the computation. When comparing the figure above to Fig. 15, note that the zero frequency is not plotted in Fig. 16. The addition of ellipsometry data (not available at press time) should allow us to correlate the change in temperature with interfacial thickness, along with allowing us to adjust for the thermal-physical effects of temperature on the plot of the power spectra. All of this is included in the complete analytical formulation we provide in Appendix B and is supported in Ref. 7.

Tables (2)

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Table 1 Surface Tension and Viscosity, Exponentially Damped Cosinea Fits

Tables Icon

Table 2 Surface Tension and Viscosity Using SRF Fitsa

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

F2=Q2 exp-σsKsεx2qsxKs-12+εxqsyKs2,
RiKs, τ=12-+-+Rζq, τF2qsx-Ks, qsydqsxdqsy,
PiKs, ω=12-+-+Pζq, ωF2qsx-Ks, qsydqsxdqsy,
Rζ=kBTγq2+gΔρexp-Γqτcosωqτ,
DˆΓ, ω, q, ρu, ρb, ηu, ηb, γ=0,
RiKs, τ=12-+-+kBTγq2+gΔρexp-Γqτ×cosωqτF2qsx-Ks, qsydqsxdqsy.
ωqγKs3ρu+ρb1/2qKs3/2=ωKqKs3/2,
ΓqΓKqKs2.
RiA, B, N, Γ, ω, τ=A -dx exp-x2×exp-Γ1-xN2τ1-xN2+α×cosω1-xN3/2τ+B.
-+exp-u2fudu=i=-nn ωinfuin+Rn.
Minimizen=1# ΛnRin-RiA, B, N, Γ, ω, τn2,
ζq*u,b0ζˆqu,bs
powerSpectrumFitUsingMannSRF=1πRepowerSpectrumAmplitude kB temperatureq2γ¯×-q2Ke+sKν-smbηb+muηu+|q|ηb+ηu-q2γ¯ν+smbρbq2-|q|mb+muρuq2-|q|mu+s|q|-mbηb+muηu-q2-ηb+ηu×2-ηb+ηu+s-ρbq2-|q|mb+ρuq2-|q|mu-q2Ke+sKν-smbηb+muηu+|q|ηb+ηu-q2γ¯+sγ¯ν+s2mbρbq2-|q|mb+muρuq2-|q|mu+s2|q|-mbηb+muηu-q2-ηb+ηu2-ηb+ηu+s-ρbq2-|q|mb+ρuq2-|q|mu-1+powerSpectrumBaselineOffset.
positiveRealRootz_ :=x+|x+iy|1/2+i Signy×-x+|x+iy|1/2/21/2,
xRez, yImz,γ¯γ+ρb-ρugq2+Beq2,  γ¯νγν+Bνq2,mbpositiveRealRootq2+iωρbηb,mupositiveRealRootq2+iωρuηu, * Comment: For a monolayer-free interface, viscoelastic coefficients we take to be zero. *siω,  Be* bending modulus *0,  Bν* bending viscosity *0,γν* transverse surface viscosity *0,Ke*=G¯+K¯, surface shear and dilational elastic modulus *0,Kν*=η¯+ζ¯, surface shear and dilational viscosity modulus *0,qsurfaceWaveNumber,ρudensityUpperPhase,  ρbdensityBottomPhase,ηuviscosityUpperPhase,  ηbviscosityBottomPhaseSought,γinterfacialTensionSought,  ggravitationalAcceleration,kB1 * Drop Boltzmann constant scale factor. *,ω2π selectedFrequencyValuesForPowerSpectrumAxis.
Gradient  {Re[δinterfacialTensionSoughtpowerSpectrumFitUsingMannSRF/.Reeqn_  1, Imeqn_  - i, Abseqn_  1],Re[δviscosityLowerPhaseSoughtpowerSpectrumFitUsingMannSRF/.Reeqn_ 1, Imeqn_  - i, Abseqn_  1], etc.}
m11m12m13m14m21m22m23m24m31m32m33m34m41m42m43m44·ζˆqu,msξˆqu,msζˆqm,bsξˆqm,bs=n11n12n13n14n21n22n23n24n31n32n33n34n41n42n43n44·ζqu,m0ξqu,m0ζqm,b0ξqm,b0.
ζq*u,m0  ξq*u,m0  ζq*m,b0  ξq*m,b0,
ζq*u,m0ζˆqu,msξq*u,m0ζˆqu,msζq*m,b0ζˆqu,msξq*m,b0ζˆqu,msζq*u,m0ξˆqu,msξq*u,m0ξˆqu,msζq*m,b0ξˆqu,msξq*m,b0ξˆqu,msζq*u,m0ζˆqm,bsξq*u,m0ζˆqm,bsζq*m,b0ζˆqm,bsξq*m,b0ζˆqm,bsζq*u,m0ξˆqm,bsξq*u,m0ξˆqm,bsζq*m,b0ξˆqm,bsξq*m,b0ξˆqm,bs=Inversem11m12m13m14m21m22m23m24m31m32m33m34m41m42m43m44·n11n12n13n14n21n22n23n24n31n32n33n34n41n42n43n44·ζq*u,m0ζqu,m0ξq*u,m0ζqu,m0ζq*m,b0ζqu,m0ξq*m,b0ζqu,m0ζq*u,m0ξqu,m0ξq*u,m0ξqu,m0ζq*m,b0ξqu,m0ξq*m,b0ξqu,m0ζq*u,m0ζqm,b0ξq*u,m0ζqm,b0ζq*m,b0ζqm,b0ξq*m,b0ζqm,b0ζq*u,m0ξqm,b0ξq*u,m0ξqm,b0ζq*m,b0ξqm,b0ξq*m,b0ξqm,b0.denom001=2q2-21+exp2|q|h0+exp2mmh0-4 exp|q|+mmh0+exp2|q|+mmh0|q|mm+-1+exp2|q|h0-1+exp2mmh0q2+m2m,m11*zetaQHatUMUpperNormalCoef2*=-q2γ¯u,m+s n11,m12*xiQHatUMUpperNormalCoef2*=s n12,m13*zetaQHatMBUpperNormalCoef2*=vanDerWaalsTerm+s n13,m14*xiQHatMBUpperNormalCoef2*=s n14,n11*zetaQUMUpperNormalCoef2*=--2smm-1+exp2|q|h01+exp2mmh0q2-1+exp2|q|h0-1+exp2mmh0|q|mmρmdenom001-sρumuq2-|q|mu+q2γ¯νu,m,n12*xiQUMUpperNormalCoef2*=iq-2ηm+2ηu+1denom001×[2s--1+exp2|q|h0×-1+exp2mmh0q2+1+exp2|q|h0+exp2mmh0-4 exp|q|+mmh0+exp2|q|+mmh0|q|mmρm+sρuq2-|q|mu,n13*zetaQMBUpperNormalCoef2*=8 exp|q|+mmh0s|q|mm-q sinhqh0+mmsinhmmh0ρmdenom001,n14*xiQMBUpperNormalCoef2*=4iexp|q|h0-expmmh0-exp2|q|+mmh0+exp|q|+2mmh0smm|q|qρmdenom001,m21*zetaQHatUMLowerNormalCoef2*=vanDerWaalsTerm+s n21=vanDerWaalsTerm+s n13,m22*xiQHatUMLowerNormalCoef2*=s n22=-s n14,m23*zetaQHatMBLowerNormalCoef2*=-q2γ¯m,b+s n23,m24*xiQHatMBLowerNormalCoef2*=s n24,n21*zetaQUMLowerNormalCoef2*=n13,n22*xiQUMLowerNormalCoef2*=-n14,n23*zetaQMBLowerNormalCoef2*=--2smm-1+exp2|q|h01+exp2mmh0q2-1+exp2|q|h0-1+exp2mmh0|q|mmρmdenom001-sρbmbq2-|q|mb+q2γ¯νm,b,n24*xiQMBLowerNormalCoef2*=iq-2ηb+2ηm+1denom0012s-1+exp2|q|h0×-1+exp2mmh0q2-1+exp2|q|h0+exp2mmh0-4 exp|q|+mmh0+exp2|q|+mmh0|q|mmρm-sρbq2-|q|mb,m31*zetaQHatUMUpperTangentCoef2*=s n31,m32*xiQHatUMUpperTangentCoef2*=-q2Ke+s n32,m33*zetaQHatMBUpperTangentCoef2*=s n33,m34*xiQHatMBUpperTangentCoef2*=s n34,n31*zetaQUMUpperTangentCoef2*=-1|q|iq-2|q|ηm+1denom0012q2-1+exp2|q|h0-1+exp2mmh0|q|-1+exp2|q|h0+exp2mmh0-4 exp|q|+mmh0+exp2|q|+mmh0mmq2-m2mηm+|q|ηu-ηumu,n32*xiQUMUpperTangentCoef2*=-q2Kν+1denom001|q|8 exp|q|+mmh0q3q2-m2m×-coshmmh0mmsinhqh0+q coshqh0sinhmmh0ηm+ηu|q|+mun33*zetaQMBUpperTangentCoef2*=8i exp|q|+mmh0q|q|coshqh0-coshmmh0mmq2-m2mηmdenom001,n34*xiQMBUpperTangentCoef2*=8 exp|q|+mmh0q3q2-m2m-mmsinhqh0+qsinhmmh0ηmdenom001|q|,m41*zetaQHatUMLowerTangentCoef2*=s n41=-s n33,m42*xiQHatUMLowerTangentCoef2*=s n42=s n34,m43*zetaQHatMBLowerTangentCoef2*=s n43,m44*xiQHatMBLowerTangentCoef2*=-q2Ke+s n44,n41*zetaQUMLowerTangentCoef2*=-n33,n42*xiQUMLowerTangentCoef2*=n34,n43*zetaQMBLowerTangentCoef2*=-iqηm+ηb-1+mb|q|+1denom0012mmηm2-1+exp2mmh0×-1+exp2|q|h0mmq2-1+exp2mmh0+exp2|q|h0-4 exp|q|+mmh0+exp2|q|+mmh0m2m+q2|q|,n44*xiQMBLowerTangentCoef2*=-q2Kν+1denom001|q|8 exp|q|+mmh0q3q2-m2m×-coshmmh0mmsinhqh0+q coshqh0sinhmmh0ηm+ηb|q|+mb.
ζq*u,m0ζqm,b0,  ζq*u,m0ζqu,m0
ζq*u,m0ξqm,b0,  ζq*u,m0ξqu,m0
ζq*u,m0ζˆqu,ms
ζq*u,m0ζˆqu,ms=Re{m22 m34 m43 n13-m22 m33 m44 n13-m14 m33 m42 n23+m13 m34 m42 n23+m14 m32 m43 n23-m12 m34 m43 n23-m13 m32 m44 n23+m12 m33 m44 n23-m14 m22 m43 n33+m13 m22 m44 n33+m14 m22 m33 n43-m13 m22 m34 n43+m24 m33 m42 n13-m32 m43 n13-m13 m42 n33+m12 m43 n33+m13 m32 n43-m12 m33 n43+m23 -m34 m42 n13+m32 m44 n13+m14 m42 n33-m12 m44 n33-m14 m32 n43+m12 m34 n43×ζq*u,m0ζqm,b0+m22 m34 m43 n11-m22 m33 m44 n11-m14 m33 m42 n21+m13 m34 m42 n21+m14 m32 m43 n21-m12 m34 m43 n21-m13 m32 m44 n21+m12 m33 m44 n21-m14 m22 m43 n31+m13 m22 m44 n31+m14 m22 m33 n41-m13 m22 m34 n41+m24 m33 m42 n11-m32 m43 n11-m13 m42 n31+m12 m43 n31+m13 m32 n41-m12 m33 n41+m23 -m34 m42 n11+m32 m44 n11+m14 m42 n31-m12 m44 n31-m14 m32 n41+m12 m34 n41ζq*u,m0ζqu,m0[-m12 m24 m33 m41+m12 m23 m34 m41+m11 m24 m33 m42-m11 m23 m34 m42+m12 m24 m31 m43-m11 m24 m32 m43-m12 m21 m34 m43+m11 m22 m34 m43+m14 -m23 m32 m41+m22 m33 m41+m23 m31 m42-m21 m33 m42-m22 m31 m43+m21 m32 m43-m12 m23 m31 m44+m11 m23 m32 m44+m12 m21 m33 m44-m11 m22 m33 m44+m13 m24 m32 m41-m22 m34 m41-m24 m31 m42+m21 m34 m42+m22 m31 m44-m21 m32 m44-1.
ζq*u,m0ζqu,m0=kBtemperatureγm,bq2+ρm-ρbg+Bem,bq4+vanDerWaalsTerm×γu,mq2ρu-ρmg+Beu,mq4+vanDerWaalsTermγm,bq2+ρm-ρbg+Bem,bq4+vanDerWaalsTerm-vanDerWaalsTerm2-1,
ζq*u,m0ζqu,m0=kB temperatureγ¯m,bq2+vanDerWaalsTermγ¯u,mq2+vanDerWaalsTermγ¯m,bq2+vanDerWaalsTerm-vanDerWaalsTerm2=kB temperatureγ¯m,bq2+vanDerWaalsTermγ¯m,bq2vanDerWaalsTerm+γ¯u,mq2γ¯m,bq2+vanDerWaalsTerm.
ζq*u,m0ζqm,b0=ζqμ,m0ζq*m,b0=kB temperature×vanDerWaalsTermγ¯m,bq2 vanDerWaalsTerm+γ¯u,mq2γ¯m,bq2+vanDerWaalsTerm,
γ¯u,mγu,m+ρu-ρmgq2+Beu,mq2,  γ¯m,bγm,b+ρm-ρbgq2+Bem,bq2,ζq*m,b0ζqm,b0=kB temperatureγ¯u,mq2+vanDerWaalsTermγ¯m,bq2 vanDerWaalsTerm+γ¯u,mq2γ¯m,bq2+vanDerWaalsTerm.
ξq*u,m0ξqu,m0=0 when no monolayer is present,ξq*u,m0ξqu,m00 when a monolayer is present.
ζq*m,b0ζˆqm,bs
vanDerWaalsTerm=-2h02HamakerConstant12πh02*The vanDerWaalsTerm function has more sophisticated forms, which are easily accommodated.** Note that an underscript u,m denotes the uppermiddle interface and that an underscript m,b denotes the middlebottom interface throughout this manuscript. *γ¯u,mγu,m+ρu-ρmgq2+Beu,mq2,  γ¯m,bγm,b+ρm-ρbgq2+Bem,bq2,γ¯νu,mγνu,m+Bνu,mq2,  γ¯νm,bγνm,b+Bνm,bq2,mbpositiveRealRootq2+iωρbηb, mmpositiveRealRootq2+iωρmηm, mupositiveRealRootq2+iωρuηu* Comment: The viscoelastic coefficients we take to be zero. *siω,  Beu,m* bending modulus *0, Bem,b* bending modulus *0,Bνu,m* bending viscosity *0, Bνm,b* bending viscosity *0,γνu,m* transverse surface viscosity *0, γνm,b* transverse surface viscosity *0,Ke*=G¯+K¯, surface shear plus dilational elastic modulus *0,Kν*=η¯+ζ¯, surface shear plus dilational viscosity modulus *0,qsurfaceWaveNumber,  h0interfacialThickness,ρudensityUpperPhase,  ρmdensityMiddlePhase,  ρbdensityBottomPhase,ηuviscosityUpperPhase,  ηmviscosityMiddlePhase,  ηbviscosityBottomPhase,γu,minterfacialTensionUpperMiddleInterface,γm,binterfacialTensionMiddleBottomInterface,ggravitationalAcceleraton,  kB1 * Drop Boltzmann constant scalefactor. *,ω2π selectedFrequencyValuesForPowerSpectrumAxis
surfaceTensionLitValue=surfaceTensionAtRefTempInKelvin×critcalTempInKelvin  tempInKelvincritcalTempInKelvin  refTempInKelvin1.4728.

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