Effect of the Nonstationary Viscous Flow in the Capillary on Oscillating Bubble and Oscillating Drop Measurements

Journal of Colloid and Interface Science 232, 25–32 (2000) doi:10.1006/jcis.2000.7181, available online at http://www.idealibrary.com on

Effect of the Nonstationary Viscous Flow in the Capillary on Oscillating Bubble and Oscillating Drop Measurements V. I. Kovalchuk,∗, † J. Kr¨agel,‡ R. Miller,‡,1 V. B. Fainerman,§ N. M. Kovalchuk,¶ E. K. Zholkovskij,∗ R. W¨ustneck,† and S. S. Dukhink ∗ Institute of Bio-colloid Chemistry, Ukrainian National Academy of Sciences, 42 Vernadsky Avenue, Kiev 252680, Ukraine; †Physics of Condensed Matter, University Potsdam, Kantstrasse 55, 14513 Teltow, Germany; ‡MPI f¨ur Kolloid- und Grenzfl¨achenforschung, Rudower Chaussee 5, D-12489 Berlin, Germany; §Institute of Technical Ecology, 25 Blvd. Shevchenko, Donetsk 340017, Ukraine; ¶Institute for Problems of Material Science, Ukrainian National Academy of Sciences, 3 Krzhizhanovsky Str., Kiev 252142, Ukraine; and kInstitute of Colloid Chemistry and Chemistry of Water, Ukrainian National Academy of Sciences, 42 Vernadsky Avenue, Kiev 252680, Ukraine Received December 28, 1999; accepted August 28, 2000

with the frequency. Nonstationarity of the oscillating flow inside the capillary should be taken into account at fast pressure changes (8–10). Effect of the nonstationarity is small when the period of oscillation is much larger than the hydrodynamic relaxation time th = aC2 /ν, where aC is the capillary radius and ν is the kinematic viscosity of gas or liquid in the capillary. In this case the velocity distribution over the capillary cross-section is almost stationary and the resistance of the capillary can be approximated by the Poiseuille law. If the period is smaller than the hydrodynamic relaxation time, the velocity profile across the capillary is nonstationary, and the pressure drop in the capillary is no longer described by the Poiseuille law. Thus, the effect of the nonstationary viscous flow in the capillary on oscillating bubble and oscillating drop measurements should be carefully analyzed. The compressibility of the medium inside the capillary should be also taken into account.

The dynamic behavior of a bubble or drop oscillating at the tip of a capillary immersed in a surfactant solution is considered. The pressure variation in the cell and the nonstationary flow in the capillary are taken into account. The amplitude- and phase-frequency characteristics of the system are obtained, which contain information about the relaxation processes at the interface and in the bulk phases. Their dependency on the system geometry, the bulk properties of contacting media, and the viscoelastic properties of the interface is analyzed. °C 2000 Academic Press Key Words: oscillating bubble dynamics; nonstationary flow; surface dilational rheology; surfactant adsorption.

INTRODUCTION

The oscillating bubble method is widely used in the study of dynamic properties of air–liquid interfaces. Different modifications of this method are shortly reviewed in (1). The complex dilatational viscoelastic modulus of the interface ε(iω) can be measured with this method as a function of frequency. It contains information about the relaxation processes in the interface after small surface area disturbances (2, 3). Harmonic bubble oscillations can be generated in a frequency interval from parts of a Hertz to some hundreds of Hertz (4). Various nonharmonic surface area disturbances can be also applied in this experiments (3–5). The study of liquid–liquid interfaces is also possible if the gas phase is replaced by a second liquid (4). To obtain correct values of the complex surface dilatational modulus ε(iω) one needs to eliminate the contributions of all hydrodynamic effects. Most important is to take into account the added mass of the liquid and the friction resistance of the capillary. In a closed measuring cell it is also necessary to consider the finite liquid compressibility (6). Inertia and viscosity effects depend strongly on the frequency. They can be neglected at small frequencies (7), but their contribution increases fast 1

DYNAMICS OF OSCILLATING BUBBLES AND DROPS

For two media with a spherical interface between them the dynamic condition holds, p1 − p2 − 2η1

∂u r1 ∂u r2 2σ + 2η2 + = 0, ∂r ∂r a

[1]

where p1 and p2 are the pressures in both phases near the interface, ∂u∂rr1 and ∂u∂rr2 are the radial derivatives of the normal velocities on the interface, η1 and η2 are the viscosities of both media, σ is interfacial tension, and a is the radius of curvature. For oscillating bubbles or drops there is one medium inside the capillary immersed into a measuring cell which contains the other medium. Usually the medium in the cell (we denote it as 1) is a surfactant solution. The medium inside the capillary (denoted as 2) can be air or any immiscible liquid (for example oil). The bubble or drop formed at the capillary tip will be considered as part of a sphere with a fixed position of the three-phase contact line (Fig. 1). The third and fourth terms in Eq. [1] are usually small because in oscillating bubble and drop experiments the

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26

KOVALCHUK ET AL.

be expressed in the form of a convolution integral (3), Zt δσ (t) =

ε(t − t 0 )δ[ln S(t 0 )] dt 0 ,

[4]

−∞

FIG. 1. Schematic of a bubble or drop oscillating at the tip of a capillary: 1, surfactant solution in the measuring cell; 2, gas or immiscible liquid in the capillary.

viscous dissipation is determined mainly by the resistance of the capillary which is usually long enough (5). Therefore the variation of the velocity derivatives over the interface can be ignored. The deviation of the pressure over the interface from its average value can be ignored, too because it is small as compared to the full pressure difference in the system. The dynamic surface tension can be obtained from Eq. (1) provided the pressures in both media near the interface are known. These pressures alter because of various hydrodynamic effects— inertia, viscous resistance, compressibility. One should take into account all these effects to obtain correct information about the dynamic surface tension. In the following, however, we will consider only small deviations from equilibrium. The equilibrium interface position is characterized by a certain meniscus height h 0 (the distance from the capillary head plane to the meniscus top, cf. Fig. 1) or by a certain radius of curvature of the meniscus a0 = a2C ( ahC0 + ahC0 ). At equilibrium we have 2σ0 = 0, p01 − p02 + a0

[2]

where p01 and p02 are the equilibrium pressures in both phases and σ0 is the equilibrium surface tension. A small deviation of the interface from the equilibrium state produces small pressure variations in both phases δp1 and δp2 , a small variation in surface area δS, and a small surface tension variation δσ . It follows from Eqs. [1] and [2] in the linear approximation, δp1 − δp2 − 2η1

2 ∂u r1 ∂u r2 2σ0 da0 δh + δσ = 0, + 2η2 − 2 ∂r ∂r a0 a0 dh 0 [3]

where δh is the variation of the meniscus height and δa = is the variation of the meniscus curvature radius. In the linear approximation and for the assumption of a uniform surfactant distribution along the surface, the surface tension variation can da0 δh dh 0

where δ[ln S(t 0 )] is the relative interfacial area disturbance, ε(t) is a function of time, and its Fourier image is the complex dilatational modulus ε(iω) (here and below the argument iω indicates the Fourier images of corresponding time functions). According to [4] the surface tension is determined not only by the current state of the interface but also by the previous changes of the interfacial area. The function ε(t) describes the contribution of any previous state of the system to the current surface tension. This function allows one to calculate the surface tension change with time for any given surface area disturbance S(t) (3). However, in the oscillating bubble or drop experiments the area change is unknown. It results from the pressures changes in both phases which in turn depend on the area change. A convenient way to find the variation of all the values with time is to use the Laplace-Fourier transformation procedure. One should apply the Laplace-Fourier transformation to Eq. [3] and express all terms via the meniscus curvature radius variation δa or the meniscus height variation δh. From [4] the surface tension variation can be expressed with the help of the complex dilatational modulus ε(iω) and the dependency of surface area on meniscus height, δσ (iω) = ε(iω)δ[ln S(iω)] = ε(iω)

d ln S0 δh(iω), dh 0

[5]

where δσ (iω), δ[ln S(iω)], and δh(iω) are the Fourier images of the functions δσ (t), and δ[ln S(t)], and δh(t), and S0 = π (h 20 + aC2 ) is the surface area at equilibrium. Pressure Variation in the Measuring Cell The pressure variation in the measuring cell near the interface δp1 is determined by inertia and compressibility of the liquid inside the cell and depends strongly on type and geometry of the cell (6). The viscosity contribution for the pressure in the cell can usually be neglected provided that the liquid is not extremely viscous. In a closed cell the pressure near the interface depends on the cell wall’s impedance. In the general case this impedance is complex and can give an additional phase shift to the measured signal. When the cell is much smaller than the wavelength and the bubble (or drop) is much smaller than the cell then the Fourier image of the pressure variation near the interface can be written in the simple form (6), δp1 (iω) =

BC δVm (iω) − ρ1 aC χ2 ω2 δh(iω), V0

[6]

C −1 ) is an effective cell elasticity modwhere BC = B1 (1 + VB01 dV dp ∂ρ1 −1 ulus, B1 = ρ1 ( ∂ p ) and ρ1 are the bulk elasticity modulus and the density of the liquid in the cell (in the linear approximation

27

EFFECT OF NONSTATIONARY VISCOUS FLOW

they are constants), V0 is the volume of the liquid in the cell, dVC is a coefficient describing the cell volume change produced dp by the pressure change, δVm (iω) is the Fourier image of the variation of the meniscus volume, and χ2 is a dimensionless coefficient. The first term in Eq. [6] describes the pressure change resulting from the compressibility of the medium in the cell and the cell deformation while the second term describes the inertia contribution. In an open or infinitely large cell the first term is zero. The inertial coefficient for a sphere of radius aC oscillating uniformly in an unbounded liquid is ρ1 aC (1, 11). For a bubble or drop formed at the capillary tip one has to include the correction coefficient χ2 reflecting the actual geometry. This coefficient depends on the ratio h 0 /aC . It can be obtained from the solution of the corresponding hydrodynamic task. With account of the meniscus volume dependency on the meniscus height one obtains the pressure variation in the measuring cell near the interface, µ δp1 (iω) =

¶ BC dVm − ρ1 aC χ2 ω2 δh(iω), V0 dh 0

¢ π ¡ where Vm = h 0 h 20 + 3aC2 . 6

as they are much shorter than the capillary. The pressure inside the bubble or drop can be also considered as approximately uniform. The pressure variation and the velocity will be supposed small enough that a linear approximation is valid. Under these assumptions the nonstationary flow of a viscous compressible medium in a capillary can be described by the set of equations of motion and continuity in the form (8, 9)

ρ2

8ρ2 ν 4ρ2 ν ∂v + 2 v+ 2 ∂t aC aC B2

∂u r1 0 η1 dh ≈ −χ11 ∂r aC dt

[8]

2η2

∂u r2 0 η2 dh ≈ χ12 , ∂r aC dt

[9]

0 0 where χ11 and χ12 are dimensionless coefficients depending only on the ratio h/ac . These coefficients can be obtained by using the solution of the respective fluid mechanics problem. Applying the Laplace-Fourier transformation to Eqs. [8] and [9] we find

2η1

∂u r1 (iω) 0 η1 ≈ −iωχ11 δh(iω) ∂r aC

[10]

2η2

∂u r2 (iω) 0 η2 δh(iω). ≈ iωχ12 ∂r aC

[11]

Nonstationary Flow of a Viscous Compressible Medium in the Capillary The pressure inside the bubble or drop δp2 depends on the pressure difference between the capillary ends. The pressure somoothening within the capillary cross-section is much faster than that along the capillary when its length l is much larger than the radius (l À aC ). In this case the pressure can be approximately considered depending only on the coordinate x parallel to the capillary axis and the local velocity has only one component directed along the capillary. The entrance regions will be ignored

∂(δp) ∂v =− , ∂x ∂t

 f (iω) =

2η1

0

∂(δp) ∂v f (t − t 0 ) dt 0 = − ∂t 0 ∂x

[12]

[13]

where B2 and ρ2 are the bulk elasticity modulus and the density of the medium in the capillary, δp = δp(x, t) is the pressure disturbance, v = v(x, t) is the velocity averaged over the capillary cross-section, f (t) is a function of time which describes the relaxation of the viscous stress in the capillary. Its Fourier image is

[7]

Neglecting the nonuniform distribution of the normal velocity derivatives at the interface we can express them approximately through the velocity of the meniscus height growth,

Zt

2  aC iω 4

r

 ¡ q iω ¢ I iω 1 aC ν q − 1, ν I ¡a iω ¢ 2

C

[14]

ν

where I1 (x) and I2 (x) are modified Bessel functions of the first and second order. Equation [12] assumes a local link between the velocity and the shear stress: local velocity distribution across the capillary determines the shear stress at the capillary wall in the given cross section. When the pressure changes very slowly the velocity distribution in the capillary has time to establish a parabolic profile, the shear stress is nearly stationary, the first and third terms in the left-hand side of Eq. [12] are negligibly small, and the resistance of the capillary is described by the second term. Then a quasi-stationary approximation can be used. For fast pressure changes one needs to take into account the nonstationary velocity distribution in the cross section of the capillary. The first and third terms on the left-hand side of Eq. [12] become significant. The third term has also a convolution structure and reflects the relaxation of the velocity distribution and the viscous stress following the pressure change at each portion of the capillary. Applying the Fourier transformation to Eqs. [12] and [13] with zero initial conditions (v = 0 and δp = 0 at t = 0) one obtains the set of equations for the Fourier images v(iω) and δp(iω), ¡ q ¢ I0 aC iω d[δp(iω)] ν iωρ2 v(iω) ¡ q ¢ = − dx iω I 2 aC ν

[15]

dv(iω) iω = − δp(iω), dx B2

[16]

where I0 (x) is the modified Bessel function of zero order. Equation [15] is a well-known equation for an oscillating flux in a

28

KOVALCHUK ET AL.

capillary (8–10). It is applicable for both compressible and incompressible fluids. For incompressible fluids the flow velocity are constant along the v(iω) and the pressure gradient d[δp(iω)] dx capillary (in respect to the coordinate x) and the flow is described only by Eq. [15] without the continuity Eq. [16] (10). After elimination of the velocity from the set of Eqs. [15]– [16], one obtains

From Eqs. [21] and [23] the variation of the pressure inside the bubble (drop) δp2 (iω) can be expressed via the meniscus height variation and the variation of the pressure in the reservoir, δp2 (iω) =

B2 β tanh(βl) dVm δp0 (iω) − δh(iω). cosh(βl) dh 0 πaC2

[24]

Quasi-stationary Approximation d2 [δp(iω)] − β 2 δp(iω) = 0, dx 2

[17]

Equation [18] can be simplified when the quasi-stationary approximation (ω → 0) holds

where ¡ q ¢ 2 I0 aC iω ρ2 ω ν q ¢. β2 = − ¡ B2 I a iω 2 C ν

β2 ≈

sinh βx sinh β(l − x) δp(iω) = δp2 (iω) + δp0 (iω) , [19] sinh βl sinh βl where δp2 (iω) is the Fourier image of the pressure variation inside the bubble or drop (at x = l) and δp0 (iω) that of the pressure variation inside the gas (liquid) reservoir connected to the opposite end of the capillary (at x = 0). The solution for the averaged velocity in the capillary is · ¸ cosh βx cosh β(l − x) iω δp2 (iω) − δp0 (iω) B2 β sinh βl sinh βl [20] and hence the Fourier image of the velocity at the capillary outlet (x = l) becomes vl (iω) = −

iω [δp2 (iω) cosh βl − δp0 (iω)]. B2 β sinh βl

dVm δh(iω) = πaC2 vl (iω). dh 0

f ¿

1 , th

[26]

a2

where th = νC is a hydrodynamic relaxation time. For air inside the capillary and a capillary radius aC ≈ 0.01 cm we have the condition f ¿ 1500 Hz. For water this frequency limit is much smaller, f ¿ 100 Hz. For larger frequencies we need to consider the nonstationary viscous stress in the capillary. In this case the hydrodynamic (aerodynamic) resistance of the capillary can be much larger and gives an additional phase shift between imposed and measured signal. Incompressibility Approximation For a large bulk elasticity modules B2 the compressibility of the medium inside the capillary can be neglected. For an incompressible medium (B2 → ∞) one can simplify Eq. [24]:

[21]

B2lβ 2 dVm δh(iω). πaC2 dh 0

[27]

This approximation is valid under the condition

[22]

Using the dependency of bubble (drop) volume on meniscus height one finds for the Fourier images at zero initial condition iω

This approximation is valid when the flow inside the capillary changes slowly enough so that the velocity distribution can establish almost a parabolic profile. This is the case at small frequencies,

δp2 (iω) ≈ δp0 (iω) −

This velocity determines the rate of the bubble (drop) volume change (the gas compressibility inside the bubble can be neglected): dVm = πaC2 vl (t). dt

[25]

[18]

The value Z (iω) = Biω2 β is the complex wave resistance of the capillary. Equation [17] describes the propagation of the direct wave and of that reflected from the meniscus. The ratio between the complex amplitudes of the direct and reflected waves depends on the meniscus impedance. The solution of Eq. [17] is

v(iω) = −

8νρ2 4ρ2 ω2 iω − . 2 3B2 B2 aC

[23]

|βl| ¿ 1,

[28]

which is valid for small frequencies. Taking into account the quasi-stationary approximation [25] from condition [28] one can find f ¿

B2 aC2 . 16π νρ2l 2

[29]

This condition gives the frequency limit f ¿ 104 Hz for air and f ¿ 4 × 106 Hz for water inside the capillary of radius

29

EFFECT OF NONSTATIONARY VISCOUS FLOW

aC = 0.01 cm and length l = 1 cm. In the capillary water (and other liquids) can be practically always considered as an incompressible medium (not in the measuring cell). Gas in the capillary can be considered as incompressible too when the capillary is not very long and narrow. When conditions [26] and [29] are fulfilled simultaneously then Eq. [24] takes the most simple form, δp2 (iω) ≈ δp0 (iω) − +

8νρ2 dVm iω δh(iω) dh 0 πaC4

4ρ2lω2 dVm δh(iω). 3πaC2 dh 0

[30]

because the capillary length is much larger than its radius. When the capillary is filled with a gas (oscillating bubble) the inertial contribution is determined by the added mass ρ1 aC χ2 of the liquid in the cell (the third term) due to the large difference of densities of the liquid and gas. In both cases the friction coefficient is determined by the medium in the capillary (imaginary part of the fifth term) due to the large capillary length and small radius (usually of the order of 1 cm and 100 µm, respectively) (5). For a quasi-stationary flow of an incompressible medium in the capillary we have according to [30] ·

¸ 2ε(iω) d ln S0 δh(iω) ≈ δp0 (iω), G 0 − ω G 2 + iωG 1 + a0 dh 0 2

[33]

The inverse transformation of this equation gives δp2 (t) ≈ δp0 (t) −

8νρ2l 4ρ2l dv v(t) − , 3 dt aC2

[31]

where according to [22] and [23] the mean velocity over the m dh . The second capillary cross section is v(t) = vl (t) = πa1 2 dV dh 0 dt C term on the right-hand side of Eq. [31] is the viscous resistance of the capillary in the quasi-stationary approximation. The last term on the right-hand side is the inertial term which includes the first correction to the quasi-stationary resistance. Therefore the numerical coefficient before this term has the value of 4/3 instead of 1.0 (8–10). AMPLITUDE-FREQUENCY AND PHASE-ANGLEFREQUENCY CHARACTERISTICS

Substituting all the contributions [5], [7], [10], [11], and [24] in the Eq. [3] one obtains the equation which describes the dependency of the meniscus displacement on the external pressure variation · 0 2σ0 da0 χ 0 η1 + χ12 η2 BC dVm − 2 − ρ1 aC χ2 ω2 + iω 11 V0 dh 0 aC a0 dh 0 ¸ B2 β tanh(βl) dVm 2ε(iω) d ln S0 + + δh(iω) dh 0 a0 dh 0 πaC2 =

δp0 (iω) . cosh(βl)

[32]

The first two terms on the left-hand side describe the elastic contribution, the third term and the real part of the fifth term describe the inertial contribution, whereas the fourth term and the imaginary part of the fifth term describe the viscous resistance. The sixth term is the contribution of the surface to both the elasticity (real part) and the viscous dissipation in the system (imaginary part). The elastic contribution of the cell (the first term) disappears for an open cell or a cell with an infinitely large volume V0 (6). When we have a system with a liquid in the capillary (oscillating drop) then the inertial contribution is determined mainly by this liquid (i.e., by the real part of the fifth term)

where G0 =

BC dVm 2σ0 da0 − 2 V0 dh 0 a0 dh 0

[34]

G1 =

0 0 8η2l dVm χ11 η1 + χ12 η2 + 4 dh aC πaC 0

[35]

G 2 = ρ1 aC χ2 +

4ρ2l dVm . 3πaC2 dh 0

[36]

The coefficients in Eq. (33) can be calculated theoretically or obtained via a special experimental calibration procedure as proposed in (12). The complex dilatational modulus can be represented through its real and imaginary parts ε(iω) = εS (ω) + iωηS (ω), where εS (ω), ηS (ω) are the dilatational elasticity and viscosity, respectively (13). The inverse transformation of Eqs. [32] or [33] gives the dependency of the meniscus position on time for a given pressure change. For established harmonic oscillations we have δh(t) = δh 0 cos(ωt + ψ).

[37]

For a nonstationary flow of a compressible medium the amplitude δh 0 and the phase difference ψ are given by q C12 + C22 δp0 δh 0 = q H12 + ω2 H22 ψ = arctan

ωH2 C2 − arctan , C1 H1

[38]

[39]

where δp0 is the amplitude of the external pressure oscillation, C1 = Re

1 , cosh(βl)

C2 = Im

1 , cosh(βl)

30

KOVALCHUK ET AL. ω0 1−h order of f 0 = 2π ≈ 3 × 103 [ (1+ ]1/2 Hz for a bubble oscillath˜ 2 )2 ing in an open cell for aC ∼ 100 µm and water–air system (5) 1−h˜ 2 1/2 ] Hz for an oscillating drop under and f 0 ≈ 3 × 102 [ (1+ h˜ 2 )3 the same conditions and a capillary length of l ∼ 1 cm. The friction in the system is described by the coefficient G 1 . It determines the damping time of a nonestablished oscillation and the phase shift of established oscillations relative to the imposed external signal. The dimensionless damping coefficient λ = ωG0 G1 2 is about 0.036 for an oscillating bubble and about 0.3 for oscillating drops (for h˜ = 0 and all other conditions given above). The discussion about the role and magnitudes of different system parameters can be found also elsewhere (5, 6). The presence of surfactants leads to a change of the characteristic frequency and to an increase of the friction (see below). It follows from Eqs. [38]–[41] that the characteristic frequency and friction coefficient should be corrected also with regard to the nonstationary flow and compressibility of the medium in the capillary. So far it was considered that the meniscus oscillations are generated by a pressure variation inside the gas (liquid) reservoir connected to the opposite end of the capillary. There is also the possibility of generating oscillations from the other side of the meniscus—via variations of the measuring cell volume keeping the pressure in the gas (liquid) reservoir constant (1, 6). In this case we obtain the same equations of meniscus oscillations with the only correction that the right-hand side of Eq. [32] should be replaced by BVC0 δVC (iω) where δVC (iω) is the measuring cell volume variation (6). Dilatational elasticity and viscosity depend on the character of the relaxation processes at the interface. Here we will consider the case of pure diffusion relaxation q of the surface tension and is much smaller than the assume that the diffusion length D ω curvature radius a0 . In this case the surface dilatational elasticity and the surface dilatational viscosity are given by the following relationships (2, 13): ˜2

and H1 =

BC dVm 2σ0 da0 B2 3R dVm − 2 − ρ1 aC χ2 ω2 + V0 dh 0 a0 dh 0 πaC2 dh 0 +

H2 =

2εS (ω) d ln S0 a0 dh 0

[40]

0 0 χ11 B2 3I dVm 2ηS (ω) d ln S0 η1 + χ12 η2 + + 2 aC a0 dh 0 πaC dh 0

3R = Re[β tanh(βl)] 3I =

[41] [42]

1 Im[β tanh(βl)]. ω

[43]

Assuming a quasi-stationary flow and an incompressible medium in the capillary one can simplify the expressions for the amplitude and the phase angle of the meniscus oscillation, δh 0 = q¡

δp0

G 0 − ω2 G 2 +

¢ ¡ ¢ 2εS (ω) d ln S0 2 ln S0 2 + ω2 G 1 + 2ηaS0(ω) d dh a0 dh 0 0

¢ ln S0 ω G 1 + 2ηaS0(ω) d dh 0 −arctan ln S0 G 0 − ω2 G 2 + 2εaS 0(ω) d dh 0

[44]

¡

ψ=

.

[45]

The sets of Eqs. [38]–[43] and [44]–[45] give the amplitudefrequency and the phase-angle-frequency characteristics of the studied system. They depend on the geometry of the system, on the bulk properties of both contacting media, and on the viscoelastic properties of the interface. The surface dilatational elasticity εS (ω) and viscosity ηS (ω) describe a surface tension response to the surface area disturbance. They can be found from the experimentally measured frequency characteristics provided that all other properties of the system are known. Equations (44) and (45) √ show that the characteristic frequency is approximately ω0 = G 0 /G 2 for the system containing only pure fluids without surfactant and it is determined by the cap0 da0 , the liquid compressibility illary pressure contribution − 2σ a02 dh 0 BC dVm in the cell and the cell deformation V0 dh 0 , and the inertial coefficient G 2 . Thus, ω0 depends on the equilibrium surface tension and radius of curvature, the cell elastic properties and its volume, the mass of the liquid in the capillary (oscillating drop) or the added mass of the liquid2 in the cell (oscillating bubble). B a 1−h˜ 2 It was found (6) that ω02 = V0CGC2 [ π2 (h˜ 2 + 1) + A (1+ ], where h˜ 2 )2 0 V0 ˜ A = 4σ 4 and h = h 0 /aC . For small values of the parameter A BC aC (less than Acr = 27π/2), ω02 is always positive and stable oscillations are possible for any meniscus position, i.e., for any h 0 , whereas at A > Acr two stable and one unstable meniscus positions are possible under the same external conditions (6). The critical parameter Acr corresponds to the cell volume of about V0 = 30 cm3 for the system air–water, a capillary radius of aC = 100 µm, and an absolutely rigid cell. The characteris˜ It is of tic frequency varies within wide limits depending on h.

q D 1+ α1 2ω q , εS (ω) = ε0 D 1+ α1 2D + ω α2 ω

ηS (ω) =

1 α

q

D

ε0 2ω q ω 1+ 1 2D + α

ω

D α2 ω

,

[46] D is the diffusion coefficient of the surfactant, ε0 = is the Gibbs elasticity modulus, α = d0 , 0 is the dc adsorption, and c is the surfactant bulk concentration. λ = √GG 1G , γ = the dimensionless variables ω˜ = ωω0 ,q 0 2 p 4ρ2 l dVm 2ε0 d ln S0 , τ = th ω0 , ζ = 3πa , κ = lω0 ρB22 , ξ = α ωD0 , 2 a0 G 0 dh 0 dh G 0 2 C one can rewrite the Eqs. [38]–[39] in the form

where − d dσ ln 0 Gibbs Using

δh 0 (ω) ˜

p δh 0 (0) C12 + C22 ¢2 ¡ 3ζ ˜ 1 − ω˜ 2 + γ ε˜ S + ζ ω˜ 2 + 4κ 3R + ω˜ 2 λ + γ η˜ S −

= q¡

6ζ τ

+

¢ 3ζ ˜ 2 3 4κ I [47]

EFFECT OF NONSTATIONARY VISCOUS FLOW

FIG. 2. The amplitude-frequency characteristics of an oscillating bubble at γ = 5 and ξ = 0.1 (curve 1), 1 (curve 2), and 10 (curve 3). Other conditions are given in the text.

31

FIG. 4. The amplitude-frequency characteristics of an oscillating drop at γ = 5 and ξ = 0.1 (curve 1), 1 (curve 2), and 10 (curve 3). Other conditions are given in the text.

When the oscillations are excited from the side of the measuring cell then Eqs. [47] and [48] will be the same with C1 = 1, C2 = 0, and δh 0 (0) = VB0 GC 0 δV0 , where δV0 is the amplitude of the cell volume oscillation.

The examples of amplitude-frequency and phase-anglefrequency dependencies for the particular case of a diffusioncontrolled adsorption are shown in Figs. 2–5. The continuous lines correspond to the quasi-stationary flow of incompressible medium in the capillary [44]–[45]. The dashed lines describe the solution where the flow nonstationarity and medium compressibility [38]–[43] are considered when the excitation is applied from the measuring cell side. The dotted lines in Figs. 2 and 3 correspond to the case of nonstationary flow of an compressible medium as well but with the excitation from the gas reservoir at the opposite end of the capillary. Figures 2 and 3 are plotted for the parameters λ = 3.6 × 10−2 , τ = 13.2, κ = 0.6, and ζ = 0.08, which illustrates the case of air in the capillary (the oscillating bubble) of radius 0.01 cm and length 1 cm. The next two figures are obtained for the parameters λ = 0.3, τ = 20, κ = 0.013, ζ = 0.985, which corresponds to the case of a second liquid-like water in the capillary (the oscillating drop) having the same radius and length as before. It is assumed

FIG. 3. The phase-angle-frequency characteristics of an oscillating bubble at γ = 5 and ξ = 0.1 (curve 1), 1 (curve 2), and 10 (curve 3). Other conditions are given in the text.

FIG. 5. The phase-angle-frequency characteristics of an oscillating drop at γ = 5 and ξ = 0.1 (curve 1), 1 (curve 2), and 10 (curve 3). Other conditions are given in the text.

¡

ψ = arctan

ω˜ λ + γ η˜ S − C2 − arctan C1 1 − ω˜ 2 + γ ε˜ S +

¢

3ζ ˜ + 4κ 31 , 3ζ ˜ 2 ζ ω˜ + 4κ 3R

6ζ τ

[48]

0 is the zero frequency limit of the amwhere δh 0 (0) = δp G0 ˜ 3 ˜ I = 1 Im[β˜ tanh(κ β)], ˜ C1 = ˜ plitude, 3R = Re[β˜ tanh(κ β)], ω˜ √ I ( iτ ω) ˜ 1 1 ˜2 ˜ 2 I0 (√iτ ω) , Re cosh(κ β) ˜ , C 2 = Im cosh(κ β) ˜ , β = −ω ˜ 2

ε˜ S (ω) ˜ =

√1 1 + ξ √12ω˜ 1 ξ 2ω˜ q q , and η˜ S (ω) = . ω˜ 1 + 1 2 + 1 1 + ξ1 ω2˜ + ξ 21ω˜ ξ ω˜ ξ 2 ω˜

[49]

32

KOVALCHUK ET AL.

in the latter case that the surfactant is soluble only in one of the liquids and that the model of diffusion controlled adsorption is applicable also in this case. The dimensionless parameter ξ is determined by the ratio , which depends on the type of of the characteristic length d0 dc the surfactant and on its bulk concentration, and the diffusion length at ω = ω0 . It varies within wide limits depending on the surfactant properties and on the characteristic frequency. This parameter strongly influences the shape of the amplitude- and phase-frequency dependencies of the system. The increase of the parameter ξ (higher surface activity) leads to an increased resonance frequency because of the higher surface elasticity. A sharp decrease of the resonance amplitude is observed at moderate values of the parameter ξ due to the maximum of the surface viscosity (see additionally (5)). The magnitude of the dimensionless parameter γ can also be very different depending on the surfactant concentration and on the equilibrium meniscus height. For an open cell we obtain ˜2 0 · 1−h h˜ 2 . The Gibbs elasticity ε0 increases with concenγ = 2ε σ0 tration from zero to some hundreds mN/m. Thus, the parameter γ changes with concentration and the meniscus height over some decades, theoretically from zero to infinity (but practically h˜ cannot be very close to unity). The effect of surfactant on the amplitude- and phase-frequency characteristics increases with γ . Figures 2–5 show that the effect of the flow nonstationarity and medium compressibility in the capillary increases with frequency. In particular, the shift of the resonance frequency and depression of the resonance amplitude is observed due to this effect as discussed above. The flow nonstationarity and medium compressibility can be neglected at small frequencies corresponding to requirements [26] and [29] but should be taken into account at large frequencies. The curves for an oscillating bubble (Fig. 2) demonstrate a second maximum which is explained through the effect of gas flow oscillations inside the capillary (fast pressure oscillations (14)) and corresponds to the first characteristic frequency of this oscillation. This oscillation is a result of the interaction of the direct wave and the one reflected from the meniscus, and appears because of the difference of the complex wave resistance of the capillary Z (iω) and the meniscus (see above). Unfortunately it is hard to observe this fast oscillation experimentally because the oscillating bubble (drop) method has a limitation in respect to the frequency following from the assumption about a nearly spherical shape of the interface. At large frequencies the different parts of the interface begin to oscillate with different phase lag and the interface is no longer part of a sphere. CONCLUSIONS

The analysis of the dynamic behavior of an oscillating bubble or drop allows one to derive the amplitude- and phase-frequency characteristics which take into account the dependency on the viscoelastic properties of the fluid–fluid interface. This depen-

dency manifests itself through the frequency functions εS (ω) and ηS (ω), characterizing the relaxation processes in the interface due to small disturbance. These functions, extracted from the experimentally measured bubble or drop frequency characteristics, can be compared with that following from theoretical models or obtained by other experimental methods. As an example the surface dilatational elasticity and viscosity following from a diffusion model of surfactant adsorption kinetics was considered. Other models including different relaxation processes can be also considered. The bubble (drop) amplitude- and phase-frequency characteristics are complex characteristics of the system. They contain information about surface relaxations as well as relaxation processes in the contacting fluids, for example, the viscous stress relaxation in the capillary which is described by a convolution integral as well. Thus, the contributions of the bulk phases to the bubble (drop) dynamic characteristics should also be analyzed. The analysis of the flow in the capillary shows that for a liquid the flow nonstationarity and for a gas its compressibility can be significant phenomena at large frequencies. The contribution of the bulk depends on both the system geometry (capillary radius and length, cell volume, equilibrium meniscus height) and the properties of the fluid phases (density, viscosity, bulk elasticity). This dependency can be expressed through some parameters which can be calculated theoretically or obtained by special calibration experiments. ACKNOWLEDGMENTS The work was financially supported by projects of the European Space Agency, the DFG (Mi418/9-1, Wu187/8-1, Wu187/6-1).

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