Abstract

Measuring the surface response function of a fluid allows us to ascertain many of its properties. Simplified surface response functions are presented for several interface conditions, including (a) a thin-film between two fluids of infinite extent, (b) the newly derived fluid–fluid interface between finite boundaries, and (c) the traditional fluid–fluid interface between infinite boundaries. The finite-boundary derivation indicates that wall effects are very short range. This portends that the effects of external vibrations, which traditionally make this measurement challenging, can be mitigated by scattering from thin fluid layers.

© 2006 Optical Society of America

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References

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  1. W. V. Meyer, G. H. Wegdam, D. Fenistein, and J. A. Mann, Jr., "Advances in surface-light-scattering instrumentation and analysis: noninvasive measuring of surface tension, viscosity, and other interfacial parameters," Appl. Opt. 40, 4113-4133 (2001).
    [Crossref]
  2. J. A. Mann, Jr., P. D. Crouser, and W. V. Meyer, "Surface fluctuation spectroscopy by surface light scattering spectroscopy," Appl. Opt. 40, 4092-4112 (2001).
    [Crossref]
  3. W. V. Meyer, "Volume and interface studies of complex liquid media," Ph.D. thesis (University of Amsterdam, The Netherlands, 2002), includes a 129 Mbyte CD-ROM of derivations, a volume and surface light-scattering data-analysis program, data sets, 3-D plots, etc.)
  4. D. Fenistein, G. H. Wegdam, W. V. Meyer, and J. A. Mann, Jr., "Capillary waves on an asymmetric liquid film of pentane on water," Appl. Opt. 40, 4134-4139 (2001).
    [Crossref]
  5. D. Langevin, ed., Light Scattering by Liquid Surfaces and Complementary Techniques, 1st ed., Vol. 41 of Surfactant Science Series, M.J.Schick and F.M.Fowkes, ed. (Marcel Dekker, 1992).

2002 (1)

W. V. Meyer, "Volume and interface studies of complex liquid media," Ph.D. thesis (University of Amsterdam, The Netherlands, 2002), includes a 129 Mbyte CD-ROM of derivations, a volume and surface light-scattering data-analysis program, data sets, 3-D plots, etc.)

2001 (3)

Crouser, P. D.

Fenistein, D.

Langevin, D.

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

Mann, J. A.

Meyer, W. V.

Wegdam, G. H.

Appl. Opt. (3)

Other (2)

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

W. V. Meyer, "Volume and interface studies of complex liquid media," Ph.D. thesis (University of Amsterdam, The Netherlands, 2002), includes a 129 Mbyte CD-ROM of derivations, a volume and surface light-scattering data-analysis program, data sets, 3-D plots, etc.)

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Figures (7)

Fig. 1
Fig. 1

Power-spectra measurements of a thin layer of pentane on water in which the layer thickness decreases with increasing temperature. These data were taken using total internal reflection (TIR) from below and plotted parametrically as functions of frequency and temperature, with the temperature-axis data being most concentrated near the bifurcation point. Compare with the theory plot shown in Fig. 2, noting that the theory plot does not include instrument function broadening. Gaps in the experimentally measured power spectra are visible where one expects a strong low-frequency contribution, e.g., at 45 ° C . Where gaps in the power spectrum exist across a complete frequency spectrum is likely a beneficial result of the use of cross correlation (which rejects signal excursions off the detector). Roll-off filters were set for 6   dB below 100 Hz and above 1   MHz .

Fig. 2
Fig. 2

Calculated power spectra for light scattered from the liquid–vapor (upper) interface of a layer of pentane on water. These power spectra are calculated without free parameters and are given as a function of the interfacial thickness of the pentane, which is temperature dependent; that is, the other material properties do not change significantly over this small temperature range and are held at 45 ° C . A Hamaker coefficient of 1.1 × 10 21   J and a grating value of q = 1000 cm 1 are used for all 200 spectra. The large shift in the power spectra as the pentane layer becomes very thin is due to the inclusion of Casimir–Polder force terms.

Fig. 3
Fig. 3

Calculated power spectra for light scattered from the liquid–liquid (lower) interface of a layer of pentane on water. See Fig. 2 for additional details.

Fig. 4
Fig. 4

Figure shows the combined theoretical power spectra of Figs. 2 and 3 by displaying the power spectra of both the upper interface and lower interface (including the critical Casimir–Polder force term for a thin film of pentane on water). Note that the power spectra for the lower interface are about an order of magnitude lower in amplitude than the power spectra for the upper interface. This model does not account for the difference in the amount of light scattered from each interface due to the relative indices of refraction of the fluids or the use of TIR.

Fig. 5
Fig. 5

Figure shows the calculated power spectra at a liquid–liquid and water–pentane interface where the lower water- and upper pentane-liquid layers are in contact with a solid boundary. These power spectra are calculated without free parameters and are given as a function of the interfacial thickness of the fluids, which is temperature dependent. The other material properties do not change significantly over this small temperature range and are held at 45 ° C . The thickness of the lower and upper liquids combined is 0.11   cm , and the thickness of the lower liquid layer is shown in the plot. A grating value of q = 1000 cm 1 was used for all 46 spectra. Note that the spectrum is not shifted or broadened by the lower boundary until the distance between it and the interface is less than 10 4   m .

Fig. 6
Fig. 6

Figure shows the calculated power spectra at the liquid–vapor and water–pentane interface. The lower-water-liquid layer and upper pentane-vapor layer are contained by solid lower and upper boundaries, respectively. These power spectra are calculated without free parameters and are given as a function of the interfacial thickness of the fluids, which is temperature dependent; the other material properties do not change significantly over this small temperature range and are held at 45 ° C . The thickness of the lower and upper liquids combined is 1.0   cm , and the thickness of the lower liquid layer is shown in the plot. A grating value of q = 1000 cm 1 was used for all 46 spectra. Note that the spectrum is not shifted or broadened by the lower boundary until the distance between it and the interface is less than 10 4   m .

Fig. 7
Fig. 7

Figure shows the calculated power spectra at the liquid–vapor and pentane–pentane interface. The lower pentane-liquid layer and upper pentane-vapor layer are contained by solid lower and upper boundaries, respectively. The comments made with respect to Fig. 6 hold here as well. Note that the spectrum is not shifted or broadened by the lower boundary until the distance between it and the interface is less than 10 4   m .

Tables (1)

Tables Icon

Table 1 Reduction of M and N Matrices for a Double Interface to Those of a Single Interface when the Film Thickness Goes to Zero

Equations (62)

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ζ q u , m * [ 0 ] ζ ̂ q u , m [ s ] = amplitude × Re [ ( m 22 m 34 m 43 n 13 m 22 m 33 m 44 n 13 m 14 m 33 m 42 n 23 + m 13 m 34 m 42 n 23 + m 14 m 32 m 43 n 23 m 12 m 34 m 43 n 23 m 13 m 32 m 44 n 23 + m 12 m 33 m 44 n 23 m 14 m 22 m 43 n 33 + m 13 m 22 m 44 n 33 + m 14 m 22 m 33 n 43 m 13 m 22 m 34 n 43 + m 24 ( m 33 m 42 n 13 m 32 m 43 n 13 m 13 m 42 n 33 + m 12 m 43 n 33 + m 13 m 32 n 43 m 12 m 33 n 43 ) + m 23 ( m 34 m 42 n 13 + m 32 m 44 n 13 + m 14 m 42 m 33 m 12 m 44 n 33 m 14 m 32 n 43 + m 12 m 34 n 43 ) ) ζ q m , b * [ 0 ] ζ q m , b [ 0 ] + ( m 22 m 34 m 43 n 11 m 22 m 33 m 44 n 11 m 14 m 33 m 42 n 21 + m 13 m 34 m 42 n 21 + m 14 m 32 m 43 n 21 m 12 m 34 m 43 n 21 m 13 m 32 m 44 n 21 + m 12 m 33 m 44 n 21 m 14 m 22 m 43 n 31 + m 13 m 22 m 44 n 31 + m 14 m 22 m 33 n 41 m 13 m 22 m 34 n 41 + m 24 ( m 33 m 42 n 11 m 32 m 43 n 11 m 13 m 42 n 31 + m 12 m 43 n 31 + m 13 m 32 n 41 m 12 m 33 n 41 ) m 23 ( m 34 m 42 n 11 + m 32 m 44 n 11 + m 14 m 42 n 31 m 12 m 44 n 31 m 14 m 32 n 41 + m 12 m 34 n 41 ) ) ζ q m , b * [ 0 ] ζ q m , m [ 0 ] ( m 12 m 24 m 33 m 41 + m 12 m 23 m 34 m 41 + m 11 m 24 m 33 m 42 m 11 m 23 m 34 m 42 + m 12 m 24 m 31 m 43 m 11 m 24 m 32 m 43 m 12 m 21 m 34 m 43 + m 11 m 22 m 34 m 43 + m 14 ( m 23 m 32 m 41 + m 22 m 33 m 41 + m 23 m 31 m 42 m 21 m 33 m 42 m 22 m 31 m 43 + m 21 m 32 m 43 ) m 12 m 23 m 31 m 44 + m 11 m 23 m 32 m 44 + m 12 m 21 m 33 m 44 m 11 m 22 m 33 m 44 + m 13 ( m 24 m 32 m 41 m 22 m 34 m 41 m 24 m 31 m 42 + m 21 m 34 m 42 + m 22 m 31 m 44 m 21 m 32 m 44 ) ) ] +  baseline.
ζ q * u , m [ 0 ] ζ q u , m [ 0 ] = k B   temperature ( γ ̄ m , b q 2 + van der   Waals   term ) ( γ ̄ m , b q 2 + van der   Waals   term ) + γ ̄ u , m q 2 ( γ ¯ m , b q 2 + van der   Waals   term ) .
ζ q * u , m [ 0 ] ζ q m , b [ 0 ] = ζ q u , m [ 0 ] ζ q * m , b [ 0 ] = k B   temperature × van der   Waals   term ( γ ̄ m , b q 2 + van der   Waals   term ) + γ ¯ u , m q 2 ( γ ¯ m , b q 2 + van der   Waals   term ) ,
γ ̄ u , m γ u , m + | ρ u ρ m | g q 2 + B e u , m q 2 , γ ̄ m , b γ m , b + | ρ m ρ b | g q 2 + B e m , b q 2 ,
ζ q * m , b [ 0 ] ζ q m , b [ 0 ] = k B   temperature ( γ ¯ u , m q 2 + van der   Waals   term ) ( γ ¯ m , b q 2 + van der   Waals   term ) + γ ¯ u , m q 2 ( γ ¯ m , b q 2 + van der   Waals   term ) .
ξ q u , m * [ 0 ] ξ q u , m [ 0 ] = 0   when   no   monolayer   is   present ,
ξ q u , m * [ 0 ] ξ q u , m [ 0 ] 0   when   a   monolayer   is   present ,
ζ q m , b * [ 0 ] ζ ̂ q m , b [ s ] = amplitude × Re ( m 21 m 34 m 43 n 12 m 21 m 33 m 44 n 12 m 14 m 33 m 41 n 22 + m 13 m 34 m 41 n 22 + m 14 m 31 m 43 n 22 m 11 m 34 m 43 n 22 m 13 m 31 m 44 n 22 + m 11 m 33 m 44 n 22 m 14 m 21 m 43 n 32 + m 13 m 21 m 44 n 32 + m 14 m 21 m 33 n 42 m 13 m 21 m 34 n 42 + m 24 ( m 33 m 41 n 12 m 31 m 43 n 12 m 13 m 41 n 32 + m 11 m 43 n 32 + m 13 m 31 n 42 m 11 m 33 n 42 ) + m 23 ( - m 34 m 41 n 12 + m 31 m 44 n 12 + m 14 m 41 n 32 m 11 m 44 n 32 m 14 m 31 n 42 + m 11 m 34 n 42 ) ) ζ q m , b * [ 0 ] ζ q m , b [ 0 ] ( m 21 m 34 m 43 n 11 m 21 m 33 m 44 n 11 m 14 m 33 m 41 n 21 + m 13 m 34 m 41 n 21 + m 14 m 31 m 43 n 21 m 11 m 34 m 43 n 21 m 13 m 31 m 44 n 21 + m 11 m 33 m 44 n 21 m 14 m 21 m 43 n 31 + m 13 m 21 m 44 n 31 + m 14 m 21 m 33 n 41 m 13 m 21 m 34 n 41 + m 24 ( m 33 m 41 n 11 m 31 m 43 n 11 m 13 m 41 n 31 + m 11 m 43 n 31 + m 13 m 31 n 41 m 11 m 33 m 41 ) + m 23 ( m 34 m 41 n 11 + m 31 m 44 n 11 + m 14 m 41 n 31 m 11 m 44 n 31 m 14 m 31 n 41 + m 11 m 34 n 41 ) ) ζ q m , b * [ 0 ] ζ q m , b [ 0 ] ( m 12 m 24 m 33 m 41 + m 12 m 23 m 34 m 41 + m 11 m 24 m 33 m 42 m 11 m 23 m 34 m 42 + m 12 m 24 m 31 m 43 m 11 m 24 m 32 m 43 m 12 m 21 m 34 m 43 + m 11 m 22 m 34 m 43 + m 14 ( m 23 m 32 m 41 + m 22 m 33 m 41 + m 23 m 31 m 42 m 21 m 33 m 42 m 22 m 31 m 43 + m 21 m 32 m 43 ) m 12 m 23 m 31 m 44 + m 11 m 23 m 32 m 44 + m 12 m 21 m 33 m 44 m 11 m 22 m 33 m 44 + m 13 ( m 24 m 32 m 41 m 22 m 34 m 41 m 24 m 31 m 42 + m 21 m 34 m 42 + m 22 m 31 m 44 m 21 m 32 m 44 ) ) +  baseline.
denom 002 = { 2 | q | m m [ cosh ( | q | h 0 ) cosh ( m m h 0 ) 1 ] + [ sinh ( | q | h 0 ) sinh ( m m h 0 ) ] ( q 2 + m m 2 ) } ,
denom 002 = { 2 | q | m m [ cosh ( | q | h 0 ) cosh ( m m h 0 ) 1 ] + [ sinh ( | q | h 0 ) sinh ( m m h 0 ) ] ( 2 q 2 + s ρ m η m ) } ,
denom002 = { 2 ( [ exp ( z m | q | ) exp ( z m m m ) ] 2 + ( 1 + exp [ z m ( | q | + m m ) ] ) 2 ) | q | m m + [ 1 + exp ( 2 z m | q | ) ] [ 1 + exp ( 2 z m m m ) ] ( | q | 2 + m 2 m ) } ,
m 11 = ζ ̂ q u , b [ s ] = s n 11 q 2 γ ¯ u , m ,
m 12 = ξ ̂ q u , m [ s ] = s n 12 ,
m 13 = ζ ̂ q m , b [ s ] = s n 13 + van der   Waals   term ,
m 14 = ξ ̂ q m , b [ s ] = s n 14 ,
n 11 = ζ q u , m = [ 0 ] = q 2 γ ¯ v u , m η u ( | q | + m u ) m u | q | + 1 denom 002 [ sinh ( | q | h 0 ) cosh ( m m h 0 ) m m | q | cosh ( | q | h 0 ) sinh ( m m h 0 ) ] s ρ m m m ,
n 12 = ξ q u , m [ 0 ] = i [ sign ( q ) η u ( | q | m u ) 2 η m q s ρ m 2 q + sinh ( | q | h 0 ) sinh ( m m h 0 ) denom 002 s 2 ρ 2 m 2 q η m ] ,
n 13 = ζ q m , b [ 0 ] = 1 denom002 [ sinh ( q h 0 ) + m m q  sinh ( m m h 0 ) ] s ρ m m m sign ( q ) ,
n 14 = ξ q m , b [ 0 ] = i { 1 denom002 [ cosh ( | q | h 0 ) + cosh ( m m h 0 ) ] s ρ m m m sign ( q ) } ,
m 21 = ζ ̂ q u , m [ s ] = van der   Waals   term + s n 21 = s n 13 + van der   Waals   term,
m 22 = ξ ̂ q u , m [ s ] = s n 22 = s n 14 ,
m 23 = ζ ̂ q m , b [ s ] = s n 23 q 2 γ ¯ m , b ,
m 24 = ξ ̂ q m , b [ s ] = s n 24 ,
n 21 = ζ q u , m [ 0 ] = n 13 ,
m 22 = ξ q u , m [ 0 ] = n 14 ,
n 23 = ζ q m , b [ 0 ] = q 2 γ ¯ v m , b η b ( | q | + m b ) m b | q | + 1 denom 002 [ sinh ( | q | h 0 ) cosh ( m m h 0 ) m m | q | cosh ( | q | h 0 ) sinh ( m m h 0 ) ] × s ρ m m m ,
n 24 = ξ q m , b [ 0 ] = - i [ sign ( q ) η b ( | q | m b ) 2 η m q s ρ m 2 q + sinh ( | q | h 0 ) sinh ( m m h 0 ) denom002 s 2 ρ m 2 2 q η m ] ,
m 31 = ζ ̂ q u , m [ s ] = s n 31 = s n 12 ,
m 32 = ξ ̂ q u , m [ s ] = s n 32 q 2 K e u , m ,
m 33 = ζ ̂ q m , b [ s ] = s n 33 = s n 14 ,
m 34 = ξ ̂ q m , b [ s ] = s n 34 ,
n 31 = ζ q u , m [ 0 ] = n 12 ,
n 32 = ξ q u , m [ 0 ] = q 2 K v u , m η u ( | q | + m u ) + 1 denom 002 [ cosh ( q h 0 ) sinh ( m m h 0 ) m m q sinh ( q h 0 ) cosh ( m m h 0 ) ] s ρ m | q | ,
n 33 = ζ q m , b [ 0 ] = n 14 ,
n 34 = ξ q m , b [ 0 ] = 1 denom002 [ m m q sinh ( q h 0 ) sinh ( m m h 0 ) ] s ρ m | q | ,
m 41 = ζ ̂ q u , m [ s ] = s n 41 = s n 14 ,
m 42 = ξ ̂ q u , m [ s ] = s n 42 = s n 34 ,
m 43 = ζ ̂ q m , b [ s ] = s n 43 = s n 24 ,
m 44 = ξ ̂ q m , b [ s ] = s n 44 q 2 K e m , b ,
n 41 = ζ q u , m [ 0 ] = n 14 ,
n 42 = ξ q u . m [ 0 ] = n 34 ,
n 43 = ζ q m , b [ 0 ] = n 24 ,
n 44 = ξ q m , b [ 0 ] = q 2 K v m , b η b ( | q | + m b ) + 1 denom 002 [ cosh ( q h 0 ) sinh ( m m h 0 ) m m q sinh ( q h 0 ) cosh ( m m h 0 ) ] s ρ m | q | .
n 14 = n 22 = n 33 = n 41 , n 34 = n 42 , n 13 = n 21 , n 12 = n 31 , n 24 = n 43.
ζ q * u , b [ 0 ] ζ ̂ q u , b [ s ] = amplitude ζ q * u , b [ 0 ] ζ q u , b [ 0 ] Re ( n 11 m 22 m 12 n 21 m 11 m 22 m 12 m 21 ) + baseline =amplitude ( k B   temperature q 2 γ ¯ u , b ) Re ( n 11 m 22 + s n 12 2 m 11 m 22 + s 2 n 12 2 ) + baseline .
P ζ [ q , ω ] = ζ q * u , b [ 0 ] ζ ^ q u , b [ s ] = Re amplitude ( k B  temperature q 2 γ u , b ) [ × ( ( - q 2 ( K e u , b + s K v u , b ) - s [ η b ( | q | - m b ) - η u ( | q | - m u ) ] ) × ( - q 2 γ ¯ v u , b - ( η b ( | q | + m b ) m b | q | + η u ( | q | + m u ) m u | q | ) ) + s [ η b ( | q | - m b ) - η u ( | q | - m u ) ] 2 ) ( ( - q 2 ( K e u , b + s K v u , b ) - s [ η b ( | q | - m b ) - η u ( | q | - m u ) ] ) × ( - q 2 ( γ ¯ i , b + s γ ¯ v u , b ) - s ( η b ( | q | + m b ) m b | q | + η u ( | q | + m u m u | q | ) ) ) + s 2 [ η b ( | q | - m b ) - η u ( | q | - m u ) ] 2 ) ] + baseline
ζ q * u , b [ 0 ] ζ ̂ q u , b [ s ] = amplitude ( k B   tempreture q 2 γ u , b + | ρ b ρ u | g ) × Re [ ( m 22 n 11 m 12 n 21 m 12 m 21 + m 11 m 22 ) ] + baseline .
m 11 = ζ ̂ q u , b [ s ] = s n 11 q 2 γ ¯ u , b ,
m 12 = ξ ̂ q u , b [ s ] = s n 12 ,
m 21 = ζ ̂ q u , b [ s ] = s n 21 ,
m 22 = ξ ̂ q u , b [ s ] = s n 22 q 2 K e , n 11 = ζ q u , b [ 0 ] = ( ( s ρ b m b q 2 | q | m b + s ρ u m u q 2 | q | m u + q 2 γ ¯ v u , b ) + ρ b m b q 2 | q | m b { 1 + 1 denom003 [ 2 ( 1 + s ) ( exp ( 2 z b m b ) [ 1 + exp ( 2 z b | q | ) ] q 2 ( exp ( 2 z b | q | ) + exp ( 2 z b m b ) 4 exp [ z b ( | q | + m b ) ] + 2 exp [ 2 z b ( | q | + m b ) ] ) | q | m b + exp ( 2 z b | q | ) [ 1 + exp ( 2 z b m b ) ] m 2 b ) ] } ρ u m u q 2 | q | m u { 1 + 1 denom004 [ 2 ( 1 + s ) ( exp ( 2 z u m u ) [ 1 + exp ( 2 z u | q | ) ] q 2 ( exp ( 2 z u | q | ) + exp ( 2 z u m u ) 4 exp [ z u ( | q | + m u ) ] + 2 exp [ 2 z u ( | q | + m u ) ] ) + exp ( 2 z u | q | ) [ 1 + exp ( 2 z u m u ) ] m 2 u | q | m u ) ] } ) ,
n 12 = ξ q u , b [ 0 ] = ( ( 2 i q ( η b η u ) + i q s ρ b q 2 | q | m b i q s ρ u q 2 | q | m u ) i q ρ b q 2 | q | m b ( 1 2 [ exp ( z b | q | ) exp ( z b m b ) ] 2 ( 1 s ) m b ( | q | + m b ) denom003 ) + i q ρ u q 2 | q | m u ( 1 2 [ exp ( z u | q | ) exp ( z u m u ) ] 2 ( 1 s ) m u ( | q | + m u ) denom004 ) ) ,
n 21 = ζ q u , b [ 0 ] = ( i η b ( q sign ( q ) m b 2 [ exp ( z b | q | ) exp ( z b m b ) ] 2   sign ( q ) m b ( | q | + m b ) 2 denom003 ) i η u ( q sign ( q ) m u 2 [ exp ( z u | q | ) exp ( z u m u ) ] 2   sign ( q ) m u ( | q | + m u ) 2 denom004 ) ) ,
n 22 = ξ q u , b [ 0 ] = q 2 K v + ( η b ( | q | + m b ) + 1 denom003 ( 2 η b ( | q | + m b ) [ 1 + exp ( 2 z b m b ) ] q 2 + { 2 + exp ( 2 z b | q | ) + exp ( 2 z b m b ) 4 exp [ z b ( | q | + m b ) ] } | q | m b + [ 1 + exp ( 2 z b | q | ) m 2 b ] ) η u ( | q | + m u ) 1 denom004 ( 2 η u ( | q | + m u ) [ 1 + exp ( 2 z u m u ) ] q 2 + { 2 + exp ( 2 z u | q | ) + exp ( 2 z u m u ) 4 exp [ z u ( | q | + m u ) ] } | q | m u + [ 1 + exp ( 2 z u | q | ) ] m 2 u ) ) , .
denom003 = { 2 ( [ exp ( z b | q | ) exp ( z b m b ) ] 2 + ( 1 + exp [ z b ( | q | + m b ) ] ) 2 ) | q | m b + [ 1 + exp ( 2 z b | q | ) ] [ 1 + exp ( 2 z b m b ) ] ( | q | 2 + m 2 b ) } ,
denom004 = { 2 ( [ exp ( z u | q | ) exp ( z u m u ) ] 2 + ( 1 + exp [ z u ( | q | + m u ) ] ) 2 ) | q | m u + [ 1 + exp ( 2 z u | q | ) ] [ 1 + exp ( 2 z u m u ) ] ( | q | 2 + m 2 u ) } .
γ ¯ u , m γ u , m + | ρ u ρ m | g q 2 + 1 q 2   van der   Waals   term   + B e u , m q 2 , γ ¯ v u , m γ v u , m + B v u , m q 2 ,
γ ¯ m , b γ m , b + | ρ m ρ b | g q 2 + 1 q 2   van der   Waals   term   + B e m , b q 2 , γ ¯ v m , b γ v m , b + B v m , b q 2 ,
γ ¯ u , b γ u , b + | ρ b ρ u | g q 2 + 1 q 2   van der   Waals   term   + B e u , b q 2 , γ ¯ v u , b γ v u , b + B v q 2 ,
van der   Waals   term= 2 h 0 2 ( Hamaker   constant 12 π h 0     2 ) .
m b positive   real   root [ q 2 + s ρ b η b ] , m m positive   real   root [ q 2 + s ρ m η m ] , m u positive   real   root [ q 2 + s ρ u η u ] .
positive   real   root [ z _ ] := ( x + | x + i y | ) + i   sign [ y ] ( x + | x + i y | ) 1 / 2 2 ,

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