Anomalous Behavior of Ultra-Low-Amplitude Capillary Waves. A

May 18, 2017 - A Glimpse of the Viscoelastic Properties of Interfacial Water? ... a threshold viscoelasticity, above which it behaves like a regular v...
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Anomalous Behavior of Ultra-Low-Amplitude Capillary Waves. A Glimpse of the Viscoelastic Properties of Interfacial Water? Antonio Raudino,*,† Domenica Raciti,† and Mario Corti‡,§ †

Department of Chemical Sciences, University of Catania, Viale A. Doria 6, 95125, Catania, Italy CNR-IPCF, Viale F. Stagno d’Alcontres 37, 98158 Messina, Italy § LITA, University of Milano, Via Fratelli Cervi 93, 20090 Segrate Milano, Italy ‡

S Supporting Information *

ABSTRACT: We investigate, both theoretically and by a differential interferometric technique, the behavior of large-wavelength capillary waves (of the order of 10−4 m) selectively excited at the surface of drops and bubbles with typical eigenfrequencies of the order of 102 Hz. The resonance peaks of gas bubbles or hydrocarbon drops in water (radius less than 1 mm) highlight anomalously small dissipation in the region of ultralow (sub-nanometric) oscillation amplitudes, reaching a plateau at higher amplitudes. This is in sharp contrast to the usual oscillating systems, where an anomalous behavior holds at large amplitudes alone. Dissipation is strongly dependent on the excited vibrational modes and, in spite of remarkable numerical differences, water-vapor and water-hydrocarbon interfaces exhibit the same overall trend. A phenomenological model was developed, based on the assumption that water possesses a threshold viscoelasticity, above which it behaves like a regular viscous fluid. The well-known Deborah number was then estimated within the anomalous region and found to lie in the range of viscoelastic fluids. In agreement with previous studies of nanohydrodynamics (e.g., atomic force microscopy measurements with sub-nanometric tip motions), the present one lends support to the idea that every self-aggregating fluid exhibits yield stress behavior, including classical Newtonian fluids like water. The essential requirement is that the applied perturbation lie below a critical threshold, above which viscous behavior is recovered. Our differential interferometric technique seems particularly suitable for this type of studies, as it allows measurement of longwavelength capillary waves with sub-nanometric resolution on the oscillation amplitudes. order of 10−4 m) selectively excited at the surface of drops or bubbles. Our experiments show that these waves exhibit an anomalously small dissipation in the ultralow amplitudes region (below 10−9 m) for typical oscillation frequencies of order of 102 Hz. Interestingly, the anomalous behavior of water is observed at small and not at high oscillation amplitudes where nonlinear effects are expected to occur, leading to a variety of complex phenomena. Simple but fundamental questions will be addressed:

1. INTRODUCTION Despite continuous research in the physics of simple fluids, water still provides an endless number of new and surprising effects, both at the nanoscale and in the mesoscale. For instance, it was discovered fairly recently that water density fluctuations are much more concentrated near air (or any hydrophobic interface) than in the bulk, while just the opposite is observed for hydrophilic interfaces (see, e.g., ref 1 and references therein). Another active field, that will be discussed later to a larger extent, involves the viscoelastic properties of water, generally considered the paradigm of classical Newtonian fluids. Even in this case, interesting viscoelastic behavior mostly emerges when considering the properties of the interfacial water rather than the bulk phenomena.2−12 Notably, this problem has been widely addressed by atomic force microscopy measurements based on sub-nanometer tip motions.3,4,6−8,11,12 The investigation of water interfaces has brought an endless number of works. In most cases, the interface has been considered motionless, a picture that has been proved to be a gross approximation of real world and, more importantly, hiding numerous and fundamental effects. In this study, we will investigate by a recent interferometric technique the behavior of large wavelength capillary waves (of © 2017 American Chemical Society

• Does the hydrodynamic response of water to subnanometric excitations behave like that of Newtonian fluids? • Which macroscopic properties of interfacial oscillations are affected by the viscoelastic properties of water? • What is the difference between air−water and air− hydrocarbon interfaces? Received: March 16, 2017 Revised: May 18, 2017 Published: May 18, 2017 6439

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where the summation is over n = 1, 2... and n ≥ |m|. It can be easily proved that the smallest vibration (quadrupolar mode) occurs at n = 2. If we consider axisymmetric deformations, the ϕ-dependent components of eq 2 disappear and the sum spans over the index n alone: S(θ , t ) = ∑n Sn(t ) Pn(θ ). We first calculate the fluid velocity field near the oscillating drop (bubble) surface. This difficult problem has already been solved for what concerns both drops and bubbles, while fewer authors have addressed the problem of shape oscillations of viscoelastic drops or bubbles.21−23 Let the fluid velocity fields be υ outside the drop (bubble) and υ′ inside. In the absence of body forces, they satisfy the usual momentum equations:

Possible answers will be given by exploiting a combination of theoretical and experimental approaches as we are going to discuss.

2. A PHENOMENOLOGICAL MODEL Fluid deformation does not linearly scale with the applied stress. At ultralow stresses, caging effects due to the tight binding of the surrounding water molecules do not allow for instantaneous escape of a water molecule from its pristine cage, so the fluid deformation begins only when some critical threshold has been reached. Below such a threshold the fluid will deform but it will recover its original shape upon removal of the force. This behavior (called yield stress fluid flow) is typical of many non-Newtonian fluids (e.g., toothpaste, cement, mud, blood) that contain suspensions of structure-making particles. In this study, we experimentally observe yield stress fluid behavior in pure water (a typical Newtonian fluid) provided the external forces (the oscillatory interfacial deformations) lie below an extremely small threshold undetectable by conventional rheological techniques. Yield stress shares significant similarities with the so-called stick−slip dynamics, that is, when the sliding motion of a body over a surface only begins above a certain critical threshold (this behavior is described, e.g., by the popular Frenkel−Kontorova model of atomic friction (see, e.g., ref 13)). Another exemplum of threshold dynamics is provided by the non-ohmic transport of charged carriers at ultralow applied potentials (see, e.g., refs 14−16). The response of the carriers is dominated by intermolecular interactions at low potentials when the particles are trapped inside potential wells. At high applied potentials, particles can easily escape from the energy well, giving rise to the usual friction-controlled ohmic transport. Similarities and differences among our topic and these interesting fields will be summarized and discussed in the last section. The notion that water may exhibit viscoelastic properties is not at all new: non-Newtonian behavior has often been observed by different techniques in confined systems, like thin water films trapped at the interfaces2−8 or in strong nonequilibrium conditions, e.g., at high frequencies.9−11 Indeed, viscoelastic behavior of water has been proposed to explain several nonideal properties observed in both normal and supercritical water.17 These studies fall in the broader field of nanohydrodynamics, a growing and poorly explored issue that requires a new formulation of well-founded fluid theories, in order to take into account the granular nature of the fluid at molecular scale.18−20 Let us consider a spherical oscillating drop (bubble) of fluid embedded into another immiscible fluid. The deformed surface r of the drop (bubble) with respect to the spherical shape can be parametrized as r = (R + S(θ , ϕ , t ))eȓ

⎡∂ ⎤ ρ⎢ + (υ ·∇)⎥υ = ∇·π ⎣ ∂t ⎦

⎡∂ ⎤ ρ′⎢ + (υ′·∇)⎥υ′ = ∇·π′ ⎣ ∂t ⎦

with ρ and ρ′ being the fluid densities outside and inside the drop (bubble), respectively. The stress tensors π and π′ are defined as π = −Pδ + τ, and π′ = −P′δ + τ′, where P and P′ are the hydrostatic pressures outside and inside the drop (bubble) and δ is the Kronecker tensor. The dynamics of the fluid upon deformation is accounted for by the rheological constitutive equations of the material relating the extra stress tensor τ to the velocity field υ. Outside the drop (bubble), we conjecture that the viscoelastic properties of the fluid (water in the present study) play a role. Many viscoelastic theories are available, such as the various Oldroyd-type models. We chose the simpler Jeffreys model (see, e.g., refs 24, 25), as it makes use of a restricted number of adjustable parameters. Accordingly, the phenomenological link between the stress tensor and the fluid deformation is described by the relationship: τ + λ1

γ ̇ ≡ ∇υ + (∇υ)T

(4)

(5)

is the deformation tensor, η is the time-independent component of the fluid dynamic viscosity, while λ1 and λ2 (generally λ1 ≫ λ226) are coefficients describing the viscoelastic properties of the fluid. When λ1 = λ2 = 0, the fluid behaves as an ideal Newtonian fluid, while when λ2 ≪ λ1 → ∞ the fluid behaves as an ideal elastic solid. We seek for a time-decaying response of the stress tensor to an external perturbation: τ = T(r)e−αt, where α is a decay constant to be determined. Whence, combining this relationship with eq 4, we get: τ≡η

(1)

1 − λ 2α γ̇ 1 − λ1α

(6)

Inside the drop (bubble), the fluid viscoelasticity may be neglected, assuming Newtonian behavior:

τ ′ = η′γ ′̇

(7)

Inserting eqs 6 and 7 into eq 3 and linearizing, we get the Navier−Stokes equations ∂υ 1 = − ∇P + ηeff Δυ ∂t ρ

∑ Sn,m(t ) Ynm(θ , ϕ) n,m

⎛ ∂γ ̇ ⎞ ∂τ = η⎜γ ̇ + λ 2 ⎟ ⎝ ∂t ∂t ⎠

where

where R is the radius of the drop at rest, S(θ , ϕ , t ) is the local amplitude of the deformation that depends on the polar angles θ and ϕ, and eȓ is the unit vector along the r direction. It is convenient to expand the oscillatory deformation amplitude S(θ , ϕ , t ) in terms of spherical harmonics Ymn (θ,ϕ) = Pmn (cos θ)eimϕ: S( θ , ϕ , t ) =

(3)

(2)

∂υ′ 1 = − ∇P′ + η′Δυ′ ∂t ρ (8)

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where both the normal frequencies ωon and the damping coefficients γn are to be determined. Solution to eq 12 reads:

1−λ α

where ηeff ≡ η 1 − λ2α . Notice that the equation for a viscoelastic 1

fluid is formally identical to that of a Newtonian one, with the elastic effects being absorbed into a renormalized frequencydepending viscosity coefficient ηeff. Equation 8 must be supplemented with the continuity equations that, for incompressible fluids, read:

Sn(t ) = Sn e−αnt ;

We shall retain only linear terms of the small quantities υ, υ′, and S . Following a standard procedure (Brenn23), we eliminate the pressure field from the linearized Navier−Stokes eq 8. The resulting fourth order partial differential equations can be integrated analytically (see Supporting Information A). We seek for a solution of the above partial differential equation as the product of a space- and of a time-dependent functions. If the fluid inside the drop (bubble) is inviscid (an exact assumption in the case of bubbles), one obtains particularly simple equations; otherwise, more involved expressions can be derived following, for instance, the procedure outlined by Miller and Scriven.27 Adopting spherical r,θ,ϕ coordinates, the axisymmetric (ϕ-independent) radial and angular components of the velocity field inside the drop that remain finite when r → 0 read:23 υ′r =

∑ A′n n(n + 1)r

n−1

Pn(cos θ ) e

∑ A′n (n + 1)r n− 1

⎛ 1 ∂υ′r ∂υ′r υ′ ⎞ η′⎜ + − r⎟ ⎝ R ∂θ R ⎠ ∂θ

n

⎛ ∂υ ∂υ′ ⎞ (δP − δP′)|s − ⎜2ηeff r − 2η′ r ⎟ ⎝ ∂r ∂r ⎠

⎡

⎛

⎢⎣ ⎝ ∂Pn(cos θ) −αt × sin θ e ∂θ n

(10a)

1 ∂ R2 sin 2 θ ∂θ

(sin θ ∂∂θ ).

In writing the previous equations, we neglected surface viscosity terms. They were introduced some years ago by Scriven et al. and are rather cumbersome28,29 and are not reported here. These surface terms depend on two parameters: the shear and the dilatational viscosities. They are relevant only when the surface viscosity coefficients are higher than those in the bulk fluid. As previously said, this situation arises when considering interfaces covered by densely packed surfactants. In the present case, however, we tackle just the opposite situation since the interfacial viscosity of a fluid in contact with air, or other hydrophobic surfaces, is lower than that of the bulk phase due to the large number of freely moving vacancies.30 Using the boundary conditions 11 and 14a,b, the series representation of S(θ , t ), eq 2, and the identity Δ|sPn(cos θ) = −n(n + 1)Pn(cos θ), we obtain after some tedious but straightforward algebra the dispersion relationship:

(10b)

where is a spherical Hankel function of argument qr. The unknown constants A′n, An, and Bn appearing in eqs 10a and 10b can be calculated by imposing proper boundary conditions at the liquid interface. Once the velocity field has been calculated, we proceed further by calculating the equation of motion for the drop (bubble) deformation S(θ , t ). The fluid velocity at the interface between the two different fluids must be identical:

αn ,0 2 αn 2

(11)

where the symbol ...|s means that the function is calculated at the drop (bubble) surface. Using the series representation for S(θ , t ) given by eq 2, together with the hydrodynamics-based eq 1A of Supporting Information A, we find that the whole motion of the interface as a superposition of n independent oscillators may be described as d2 d Sn(t ) + γn Sn(t ) + (ωno)2 Sn(t ) = 0 dt dt 2

s

σ (2 + Δ|s )S(θ , t ) = 0 R2

surface operator Δ|s takes the form: Δ|s =

⎞⎤ hn(1)(qr ) ⎟⎥ − hn(1) + 1(qr )⎟ qr ⎠⎥⎦

∂S(θ , ϕ , t ) ∂t

−

where σ is the interfacial tension and δP and δP′ are the excess pressures calculated with respect to the equilibrium value Po − P o ′ given by the well-known “capillarity condition” 2σ Po − Po′ + R = 0. In spherical coordinates, the axisymmetric

h(1) n (qr)

υr |s = υr ′|s ≈

=0 s

(14b)

⎡ h (1)(qr ) ⎤ ⎥n(n + 1)Pn(cos θ) e−αt υr = − ∑ ⎢A nr −n − 2 + Bnq2 n ⎢ qr ⎥⎦ n ⎣

∑ ⎢−A nnr −n− 2 + Bnq2⎜⎜(n + 1)

s

⎛ 1 ∂υr ∂υ υ⎞ − ηeff ⎜ + r − r⎟ ⎝ R ∂θ R⎠ ∂θ

(B) normal stress balance:

Likewise, the velocity field outside the drop (bubble) and vanishing when r → ∞ is

υθ =

(13)

(14a)

−αt

∂Pn(cos θ ) sin θ e−αt ∂θ

⎥ γn 2 − 4(ωno)2 ⎦

(A) tangential stress balance:

n

υ′θ =

1⎢ γ ± 2⎣ n

In the zero-viscosity limit (γn = 0), eq 13 reduces to o Sn(t ) = Sn eiωn t + conjugated complex. In the following, we will derive explicit formulas for the normal frequency ωon and the damping coefficient γn. The first boundary condition is provided by eq 11, and other boundaries arise from the force balance. The interfacial force balance yields two boundary conditions when applied in the eȓ and eθ̑ directions; up to first order terms, we obtain

(9)

∇·υ = ∇·υ′ = 0

αn =

=

2(n + 1)(n + 2) nρ nρ + (n − 1)ρ′ q2R2 ⎡ ⎤ Q n(qR ) − (2n + 1) ⎥−1 × ⎢1 − n 1 ⎢⎣ Q n(qR ) − 2 q2R2 − (2n + 1) ⎥⎦ nρ − nρ + (n − 1)ρ′ ⎡ ⎤ n(n + 2) ⎥ ×⎢ ⎢⎣ Q (qR ) − 1 q2R2 − (2n + 1) ⎥⎦ n (15) 2

where (1) Q n(qR ) ≡ qR ·hn(1) + 1(qR )/ hn (qR )

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For inviscid fluids (η → 0), the expected result is recovered: αn → iαn,0 = iωon. For finite viscosity (Gn/ωon ≪ 2, i.e., millimeter-sized bubbles (drops) embedded in a moderately viscous fluid) and small elasticity (i.e., λ1 ≪ 1, λ2 ≈ 0), we obtain

and αn,0 is the oscillation frequency of a drop (bubble) within an ideal inviscid medium,31 ωno = Im αn ,0 =

n(n − 1)(n + 1)(n + 2) n + (n + 1)ρ′/ρ

σ ρR3

(17)

The relationship 15 is the characteristic equation for the complex angular frequency αn for a gas bubble (ρ′ ≈ 0) or an inviscid fluid drop (ρ′ ≠ 0)) oscillating inside a viscoelastic medium in the oscillation mode n. The argument qR of the Hankel function h(1) n (qr) is related to αn by ρR2 1 − λ1αn qR = αn η 1 − λ 2αn

Re(αn) ≈

Im(αn) ≈ ωno

2 2

where Gn ≡

(18)

Re(αn) ≈

qR

,

Im(αn) = 0

(19a)

where: Wn ≡

n(n + 2)2 nρ −1 nρ + (n − 1)ρ′ 2n + 1

Kn ≡

η nρ 2(n − 1)(n + 1)(n + 2) 2 nρ + (n − 1)ρ′ ρR

and Re(αn) and Im(αn) meaning the real and imaginary parts of αn. For large elasticity values (λ1 ≫ 1,λ2 ≈ 0), we find Re(αn) ≈

ωno Wn

1−

Kn λ1(ωno)2

,

Im(αn) = 0

nρ 2(2n nρ + (n + 1)ρ ′

Gn 2

4(ωno)2 λ1Gn)3

+ 1)(n +

≈ ωno η 2) 2 . ρR

(20a)

In the opposite

Gn 2(ωnoλ1)2

, Gn λ1(ωno)2

≈ ωno

(20b)

That is, for viscous bubbles (or drops) vibrating in a tough viscoelastic medium, the dissipation (Re(αn)) f irst slightly increases, then it sharply decreases by adding elasticity, while the eigen-frequency (Im(αn)) is not significantly affected. The calculations developed so far highlight the differences between a bubble (drop) vibrating in viscous and in viscoelastic media. Elastic properties considerably affect the associated energy dissipation as shown by eq 20, but, in the adopted linear approximation, they are unable to predict the narrowing of the resonant peak observed at small oscillation amplitudes. To take one step forward, we have to assume that the elastic behavior of water occurs only at the smallest perturbation, while at higher perturbations water behaves like a standard viscous medium. Such a hypothesis is supported by several experimental findings reported in the introduction2−12 and by theoretical works on the nonlinear mobility of particles trapped in a potential well,14−16 as discussed in the introduction. In other words, we suppose the water interface to behave like the so-called yieldstress fluids (see, e.g., ref 38), which exhibit viscous behavior only above a critical threshold (extremely small in the present study). In yield-stress fluids, the stress tensor and the fluid deformation are linked by the simple phenomenological relationship:38

Q n(qR ) ≈ 2n + 3 + O(q 4R4). Replacing this result into eq 15 and expanding it in power series of qR, we get a perturbation solution which, in the limit of small elasticity (λ1 ≪ 1, λ2 ≈ 0), yields an overdamped behavior: K n 2 + 4(ωno)2 (Wn − λ1K n) − K n

(1 −

Im(αn) ≈ ωno 1 +

2 2

2(ωno)2

1 − λ1Gn −

limit of large elasticity (λ1 ≫ 1, λ2 ≈ 0) we get:

which depends on both viscous (η) and elastic (λ1 and λ2) properties. The transcendental algebraic eq 15 in the unknown αn applies to both viscous and viscoelastic fluids, with the dependence on the elastic properties being contained in the argument qR (see eq 18). Equation 15 does not admit analytical solutions and compact expressions can be obtained only in a few relevant limiting cases. In a very viscous fluid qR → 0, thus we can employ the power series expansion:32

Re(αn) ≈

Gn 1 1 ≈ Gn , 2 1 − λ1Gn 2

τ < τcrit ⇒ γ ̇ = 0

(19b)

and

τ > τcrit ⇒ τ = τcrit + f (γ )̇ (21)

Qualitatively similar expressions relating the decay rate αn to the fluid viscosity η have been reported in the literature for bubbles (see, e.g., refs 33,34) or drops in a vacuum35,36 or in a viscoelastic medium.21−23 Equations 19a and 19b describe the limiting behavior of an extremely viscous medium giving rise to overdamped relaxation of the oscillating drop (bubble), that is Im(αn) = 0. Such a situation falls beyond our experimental conditions and will not be considered any longer. By lowering the friction of the medium, we foresee a different behavior. We follow the procedure sketched above in the opposite relevant limit qR → ∞. By employing the asymptotic expression for the spherical Hankel functions,37 the function Qn(qR) appearing in eq 15 and defined by eq 16 can be easily calculated: Qn(qR) ≈ −iqR + O(q−1R−1). Inserting this result into the dispersion relationship 15 we can easily find the approximate expressions for αn, as shown in Supporting Information B.

A popular expression for f(γ̇) comes out from the Herschel− Bulkley model: f(γ̇) = ηγ̇p, where η is the fluid viscosity and p is an empirical parameter. When p = 1, the Bingham equation is recovered. We will not make use of eq 21 in our model; here we want to stress the striking similarities among different theoretical approaches. In the following, we develop a simple model to tackle the nonlinear behavior of an oscillator inside a Bingham-type fluid. We treat the vibrating bubble (drop) as a linear oscillator with nonlinear friction γn submitted to an external force B(t): d2 d Sn(t ) + γn Sn(t ) + (ωno)2 Sn(t ) = B(t ) dt dt 2

(22)

The applied force B(t) undergoes periodic oscillations of frequency ω and intensity B: B(t) = Beiωt + conjugated complex. The eigen-frequencies of the n-th vibrational mode of the 6442

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Langmuir bubble (drop) in vacuum are ωon (see eq 17). As stated at the beginning of this section, the basic assumption of our model is that γn does depend on the averaged amount of energy released by the vibrating interface onto the contacting fluid. Therefore, when the energy delivered to the fluid is high, it behaves as a purely viscous medium, while at low energy supply the interfacial liquid layers yield a viscoelastic behavior. That is γn ≈ P1 + P2⟨E DISS⟩

2 max max min where f n(ω) ≡ ((ωon)2 − ω2)2 + γmax n ω , Cn ≡ 2γn (γn − γn ), min max −2 2 Dn ≡ (γn /γn )Acrit . Notice that the amplitudes An do not contain imaginary terms, as it should be.

Whenever

1 2

(23)

dSn(t ) dt

γnmin = γn(λ1max ) ≈

2

A n → Acrit

and γmin γmax n n will be derived shortly. The first identity imposes that at extremely small oscillation amplitudes, An → 0, the damping is typical of a viscoelastic medium of rigidity λ1 → max λmax is the largest elastic constant of the interfacial 1 , where λ1 region, attained at the smallest deformation An → 0. The second identity claims that beyond a critical value, say Acrit, the damping increases reaching the value γmax typical of a purely n viscous medium. Therefore, the interpolation formula 23 yields γnmin

γn ≈

1−

γnmax − γnmin |A nA n*| γnmax Acrit 2

γn = γnmax ,

,

by eqs 25a and 25b: Bcrit = ωno

A n < Acrit

(26b)

γnmax γnmin

Acrit ≡ f (n)Acrit . Using the

max o explicit expressions for γmin n , γn , and ωn (eqs 17, 26a, and 26b), 2 3/2 we calculate: f(n) ∝ (n − 1)(n + 2) (2n + 1)1/2 ≈ n4 in the limit n ≫ 1. Hence:

(24a)

A n > Acrit

Gn o max 2 (ωn λ1 )

with the coefficient Gn (defined after eq 20) being proportional to the fluid viscosity η. Numerical estimates can be obtained using the following parameters: the air−water interfacial tension σ is 7 × 10−2 Nm−1, the water density ρ and the water viscosity η are 103 kg m−3 and 10−3 kg m−1 s−1, respectively. In the case of bubbles, the internal density of the gas is ρ′ ≈ 0, while the density of hexadecane drops in water is ρ′ ≈ 0.77 × 103 kg m−3 and the hexadecane−water interfacial tension is 5.3 × 10−2 N m−1. For the sake of simplicity, the liquid drop was taken to be inviscid, while we considered the viscosity of the surrounding medium alone (excellent approximation in the numerical estimate of the frequencies). For an air bubble of radius R = 6.5 × 10−4 m, eq 15 gives the vibrational frequencies f 2 = ωo2/2π ≈ 270 Hz and f4 ≈ 750 Hz. For a hexadecane drop of R = 0.85 mm, eq 15 gives the vibrational frequencies f 2 = ωo2/2π ≈ 110 Hz and f4 ≈ 308 Hz. These are the numbers found in our experiments. The previous equations allow to estimate the position of the boundary between the viscoelastic and viscous regions defined

2π , ω

An →0

(26a)

λ1→ 0

eventually we get ⟨EDISS⟩ ≈ 2πωγn|AnA*n |. We now combine this result with eq 23, and make use of two relationships to eliminate the unknowns P1 and P2 from eq 23: lim γ n=γnmin and lim γn = γnmax . Analytical expressions for =

is not so small, additional higher order

γnmax = lim γn ≈ Gn

( ) dt . Recalling that cycle duration

∫cycle γn

γnmax

terms should be added to eq 25. The result contained in eq 25a is valid in the case of extremely small values of the external perturbation B, that is, for small oscillation amplitudes. The friction coefficients for bubble (drop) vibrating in a viscoelastic max max (γmin = γn(λ1 = 0)) medium n = γn(λ1 = λ1 )) and viscous (γn can be approximately calculated by combining eqs 13 and 20:

Where P1 and P2 are constants to be determined and ⟨EDISS⟩ is the energy dissipated per second per unit volume associated with the interface wandering: ⟨EDISS⟩ = ⟨EDISS⟩cycle(cycle duration). According to the boundary condition eq 11, the velocity of the fluid molecules at the moving interface approaches that of the interface itself. In the linear approximation, the solution to eq 22 is Sn(t ) ≈ Aneiωt + An*e−iωt, with An* being the conjugated complex. The coefficients An and A*n have to be determined. Whence, the energy dissipated per cycle turns out to be ⟨E DISS⟩cycle ≈

γnmax − γnmin

(24b)

Bcrit ∝ n 4Acrit

(26c)

Evidencing a close relationship between γn and the amplitude An. Inserting this result into eq 22 and separating the real and imaginary parts as shown in the Supporting Information C, we obtain in the limit of a small viscoelastic behavior

highlighting a sharp dependence of the critical threshold on the nth vibrational mode. In the relevant limit Cn = 0 (drop (bubble) oscillating in a purely viscous medium), eq 25a

γnmax − γnmin

reduces to eq 25b: A n 2 ≈

γnmax

A n2

≪ 1:

for extremely small B

A n2 ≈

(25a)

B2 fn (ω)

for small B

⎛ ⎞ max ⎜B > ω o γn ⎟ A n ⎜ min crit ⎟ γn ⎝ ⎠

that is, the response of the

system is of resonant type and it is described by a classical Lorentzian function. Equations 25a and 25b are the constitutive equations of our phenomenological model. Albeit approximate, they capture the most important physical effects. When the oscillation amplitude An increases further (i.e., large B values), nonlinear terms (proportional to Sn 2(t ) and Sn3(t ) and neglected in the constitutive eq 22) play a significant role and a deformation of the resonance peak from the classical Lorentzian shape emerges at high B, as described in detail by several studies on large-amplitude oscillations of drops and bubbles.39−42 Therefore, nonideal behavior should be observed at either small or large perturbations. The overall shape of the calculated resonance peak does not appreciably differ from the Lorentzian shape (compare eqs 25a

⎡ ⎤ B2 ⎢ B 2 ⎞⎥ ω2 ⎛ ⎜⎜1 − Dn ⎟⎟ 1 + Cn ≈ fn (ω) ⎢⎣ fn (ω) ⎝ fn (ω) ⎠⎥⎦ ⎛ ⎞ max ⎜B ≤ ω o γn A ⎟ n ⎜ min crit ⎟ γ ⎝ ⎠ n

B2 , f (ω)

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generated by letting air (hexadecane) pass through the internal hole of a lower electrode, then they are attached from the top to a second electrode placed above. Deformation from the spherical shape is detected by the change in the optical path inside the bubble (drop) of a laser beam which traverses it in the horizontal direction. Refractive index mismatch causes light reflection at the gas−liquid (liquid− liquid) interfaces, which act as the mirrors of a confocal Fabry−Perot interferometer. The reflected beams interfere into a set of fringes. The change from light to dark in a fringe corresponds to a change of λ/2 in the optical path, that is, to a radius change of λ/(8n), equal to 79.1/n nm for the He−Ne laser (λ = 632.8 nm), with n being the refractive index inside the bubble (drop). In the presence of a net electrical charge at the interface, a periodic electric field applies an alternating force to the drop in the vertical direction and generates capillary waves at known frequencies. The technique is differential. Lateral movements of the bubble (drop) do not affect the optical path inside it at first order. Nevertheless, lateral movements do affect the internal optical path at second order, due to hydrodynamic interactions with the external fluid. The position fluctuations of a free drop standing by buoyancy below an electrode can be as large as 10−6 m because of the acoustic noise in a typical laboratory, as we observed in a previous work.47 The differential nature of the interferometer reduces the effect of these position fluctuations to a noise level of fractions of a nanometre. Further noise reduction is achieved by minimizing the fluctuations about equilibrium position, as the oscillating bubble (drop) is pinned to the upper electrode instead of floating freely. This procedure introduces a conceptual difference with the theoretical model based on the assumption of a free oscillating bubble. However, as proved by different authors (see, e.g., 48) such differences are generally negligible. In the present work, the ultimate sensitivity of about 10−11 m in the bubble (drop) oscillation measurements was reached taking further precautions. The interferometer tightly mounted on a Newport table is surrounded by an acoustic insulating box and stands on a pneumatically stabilized table. The table legs are immersed in sand basins, thus not being in direct contact with the laboratory floor. The laboratory is placed in the basement of the building. Ultimately, the most delicate measurements on bubble oscillations in water were performed at night in order to avoid disturbances due to people walking around. These figures are far better than other interferometric detections made, for instance, by the dualwavelength reflection interference contrast microscopy.49 The “periodic” boundary conditions naturally provided by the bubble (drop) closed geometry allow only a discrete spectrum of stationary interfacial oscillation modes. The frequency response of the bubble (drop) to a sweeping sinusoidal field is measured by a multichannel look-in technique. Besides the accurate passive acoustic insulation of the apparatus mentioned above, an averaging process of repeated spectra is used to damp uncorrelated ambient noise. For the lowest amplitude measurements averages of up to 50 spectra, lasting about 10 s each, were used. Moreover, the system stability was always checked by controlling that the asymptotic large oscillation amplitudes (in the nm range) were the same at the start and at the end of the measurement run.

and 25b); however, the intensity and width of the peaks are significantly different. Three independent parameters can be derived from a careful analysis of the resonance peaks: f requency, intensity, and width. It can be easily seen that ωmax n ≈ ωon, that is, the resonant frequency approaches the eigenfrequency of the bubble (drop) in vacuum. The peak intensity L = ⟨An2⟩1/2 can be easily calculated from eqs 25a and 25b. Systematic small deviations from the linear behavior of L over the field intensity B were found. However, the most striking difference is observed when we consider the peak shape. This is best described by the Inverse quality factor: Q−1 = Δωn/ωmax n , defined as the full width at halfmaximum (Δωn) over the frequency of the maximum (ωmax n ). The explicit expression for a weakly damped driven harmonic oscillator is well-known in basic physics: Q−1 ≈ γn/ωno

(27)

a result valid for rather narrow resonant peaks. Equation 27 shows that Q−1 is strictly related to the dissipation coefficient γn, and is independent of the oscillation amplitude. No simple analytical expressions for Q−1 can be obtained for a nonlinear oscillator, therefore we adopted a numerical procedure. By employing the explicit expressions for the bandwidth as a function of the applied frequency ω (eqs 25a, 25b), we numerically calculated the parameter Q−1 using Mathematica 10.0. Some results are shown in Figure 1, where we plot Q−1

Figure 1. Schematic variations of the inverse quality factor Q−1 of the resonant peak against the intensity B of the applied field. The two dashed horizontal lines refer to a purely viscous (blue) and to a linear viscoelastic (black) model for the fluid surrounding the oscillating bubble (drop), respectively. The red curve describes the variation of Q−1 calculated by our amplitude-dependent Bingham-like viscoelastic model (eqs 25a ,25b). Beyond some critical value of B, say Bcrit, Q−1 does not depend any longer on the intensity of the field.

4. EXPERIMENTAL RESULTS Bubbles (drops) in water exhibit a discrete spectrum of vibrational modes when excited by a tunable oscillating field (either electric or acoustic). Frequencies, amplitudes and widths of the resonance peaks can be accurately measured and compared with predictions from hydrodynamics-based theories, or with computer experiments stemming from MD simulations. Of particular interest is the possibility to excite (and carefully detect) motions of ultralow amplitudes (about 0.1 nm) which may follow a more complex hydrodynamics related to the molecule-sized displacements. In Figures 2−4, we report recent measurements performed on water−air and water-hydrocarbon interfaces at temperature

versus the intensity B of the applied field. As it can be seen, Q−1 turns out to be independent of the applied intensity B both in a purely viscous system and in a linear viscoelastic medium. On the contrary, if we assume that the elastic properties of the interfacial region of the medium are modulated by its oscillation amplitude, Q−1 increases by rising the field B, reaching a plateau at high external fields.

3. EXPERIMENTAL APPROACH The employed apparatus is based on a new interferometric technique which has recently been applied to study surfactant monolayers adsorbed at the interface of a gas bubble in water.43−46 This technique was extended to immiscible drops (hexadecane in water) and used in the present work. Bubbles (drops) about 1 mm in diameter are 6444

DOI: 10.1021/acs.langmuir.7b00895 Langmuir 2017, 33, 6439−6448

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Figure 2. (A) Hexadecane drop in water. Excitation field: 2.5 V/cm, resonant frequency ωo2/2π = 107 Hz, bandwidth Δω2/2π = 48 Hz, drop radius 0,8 mm. (B) The same drop excited by a much smaller field (0.125 V/cm). Resonant frequencyωo2/2π = 103 Hz, bandwidth Δω2/2π = 25 Hz. The red full lines describe Lorentzian fits. Notice the sharp shrinking of the peak shape in (B) with respect to (A).

Figure 3. (A) Measured variation of the oscillation amplitude L = ⟨An2⟩1/2 against the intensity of the applied electric field B (semilogarithmic scale) for an air bubble in water. Red dots were calculated by assuming a strictly linear relationship between L and B. Data refer to the vibrational mode n = 2. Similar behavior was observed for the mode n = 4. (B) Anomalous dissipation behavior of ultrasmall surface vibrations. The inverse quality factor Q−1 was normalized to the value obtained in the limit of large vibration amplitudes (dashed lines). Left: a gas-filled bubble, radius 0.65 mm. Right: a hexadecane drop, radius 0.8 mm. A plateau is attained above 0.4 nm for the former system and 5 nm for the latter. The Q−1 values at large vibration amplitude for the gas-filled bubble and the hexadecane drop are 0.12 and 0.45, respectively. Yellow and violet regions mark viscoelastic and viscous regimes, respectively. Because of the quasi-linear relationship between the amplitude L and the field B reported in (A), a similar curve is obtained when plotting Q−1 against B.

of 20 °C and under external pressure of about 1000 mbar. Very pure water (bidistilled with a quartz apparatus) and chromatographic grade hexadecane were used in all the experiments. Typical exempla of the observed resonant peaks are shown in Figure 2. For amplitudes in the nanometer range, measured resonances are perfectly reproduced by a Lorentz fit; see Figure 2A as an example. Angstrom and subangstrom measurements show some scatter of the experimental points, but the Lorentz fit is still quite acceptable (Figure 2B). Figure 3A displays the intensity L of the oscillation amplitude (L = ⟨An2⟩1/2, brackets denoting time average) as a function of the strength B of the applied field for an air bubble in pure water. The red dots were calculated assuming perfect linearity between the two physical quantities, the black squares, instead, are the experimental points. Perfect linearity has always been verified for large amplitudes, for both bubbles and drops. For the bubble of Figure 3A, a small but unambiguous nonlinear behavior of L with B is observed for small amplitudes. Differences between the red dots and the black squares are well above experimental errors. Variations in the vibrational frequencies are found to be negligible. The most relevant parameter, however, is the width of the resonant peak, which is proportional to the viscous energy dissipation. In Figure 3B, the Inverse quality factor Q−1 (full width of the resonant peak measured at half height) is plotted against the oscillation amplitude of the lower vibrational mode (n = 2) for a gaseous bubble (blue) and for a hexadecane drop (red) in water. Figure 4 reports Q−1(L) for different vibrational

Figure 4. Comparison between the anomalous dissipation at ultralow excitation for the vibrational modes n = 2 (red) and n = 4 (black) of the gas bubble. The inverse quality factor Q−1 was normalized to the values obtained in the limit of large vibration amplitudes.

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properties of materials (De = 0 viscous fluids, De → ∞ elastic solids). Adopting the numerical estimates reported above, we found De to be in the range 1−10, figures often observed in common viscoelastic systems, like polymer solutions and melts or wormlike micellar solutions.52 b. Variation of the Viscoelastic Behavior along with the Normal Modes. The viscoelastic behavior of an oscillating drop or bubble strongly depends upon the kind of the excited vibrational mode n. We calculated that the resonant peak in the anomalous region where water exhibits an amplitude-related viscoelasticity is narrower than in the viscous region. The width of the viscoelastic region depends on the perturbation strength and beyond a critical value, Bcrit = f(n)Acrit, water behaves like a Newtonian fluid (see eqs 25a, 25b)). We calculated f(n) ≈ const n4 in the limit n ≫ 1. The qualitative theoretical behavior of the inverse quality factor Q−1 calculated for the modes n = 2 and n = 4 shows an overall smaller variation of Q−1 with the applied field B on increasing n because of the widening of Bcrit with n. This behavior was qualitatively confirmed by the experimental data reported in Figure 4 where we compare the dissipation of the excited modes n = 2 and n = 4. Data highlight the widening of the anomalous viscoelastic region. Consequently, the variation of Q−1 on the field B is slower on going from the mode n = 2 to n = 4. Unfortunately, the overall intensity of the resonance peaks decreases with n, making their detection difficult. Thus, no reliable data are available for large n values. c. Comparison between Different Interfaces. Our data confirm that, qualitatively, the liquid−vapor interface of water has properties similar to those of the liquid water-hydrocarbon one, in agreement with previous theoretical and experimental studies (see, e.g., refs 1, 53, 54). Interesting differences, however, are worth mentioning. Important effects are the larger damping and a downward frequency shift when we compare a hydrophobic drop oscillating in water with a gas-filled bubble suspended in the same aqueous medium. The frequency variation is captured by the model through the differences in the interfacial tension and density between bubbles. The larger damping, however, cannot be foreseen because we made the approximation of an inviscid drop. Although this approximation can be removed as shown by Scriven and Miller for immiscible viscous fluids,27 the resulting equations become intricate. In the framework of a similar model, at constant wavenumber, Lucassen and Lucassen-Reynders55 predicted damping at an interface between two equally viscous liquids to be twice as high as at air/water, where only one liquid is viscous. This result is in qualitative agreement with our experimental findings reported in Figure 3B, where we compare the damping (Q−1) of a drop of hexadecane in water with the dissipation of a gas bubble in water. Rather puzzling, however, is the explanation of the width of the viscoelastic region, which is much narrower in bubbles than in drops. At present, we have no explanation for this unexpected phenomenon. The present study lends further support to the idea that every self-aggregating fluid (including water) may exhibit yield stress, provided the applied perturbation (driven oscillation of the gas-fluid or fluid−fluid interfaces) lies below a critical threshold (in the sub-nanometric range for what concerns water). Below this threshold viscoelastic properties arise, while at higher perturbations the usual viscous behavior is recovered. Therefore, a phenomenological picture of the rheological properties of a fluid should include, at least, two additional empirical parameters: the elastic properties of the interface λmax 1

modes, n = 2 (red) and n = 4 (black), of the same gaseous bubble. Ideally, the peak width should be independent of the amplitude in the limit of linearization (harmonic oscillator), while deviations (anharmonic behavior) are expected at large amplitudes alone. Our experimental results contradict with this firm dogma: the anomalous behavior is observed right in the ultralow amplitudes regime, where drops and bubbles should display harmonic behavior. Notice that at higher amplitudes the dissipation data converge to a constant value, provided the oscillation amplitudes are much smaller than the size of the oscillator, which is the case of our experiments.

5. COMPARISON BETWEEN PREDICTED AND EXPERIMENTAL RESULTS The majority of studies conducted so far focus on planar, rigid interfaces between water and hydrophilic or hydrophobic media ranging from air to biological systems (such as proteins and membranes). Less investigated are the boundary regions between immiscible fluids where the interface undergoes periodic displacements from equilibrium position (capillary waves). We tackled this problem for water−air and waterhydrocarbon interfaces vibrating under the effect of a periodic field. For a detailed account of the main research on this topic, we refer to classical textbooks (see, e.g., refs 50, 51). Here we want to stress only a few results arising from our measurements and phenomenological theory. a. Anomalous Behavior at Subnanoscale Amplitudes. Following recent evidence concerning the viscoelastic behavior of water at the nanoscale,2−12 we hypothesize that water behaves like a viscoelastic fluid whose elastic properties decrease on increasing the amount of energy transferred from the oscillating interface into the surrounding fluid. When the transferred energy is high (namely at larger oscillation amplitudes), water behaves like a regular Newtonian fluid. This behavior is close reminiscent of the yield stress fluids flow often observed in structures-developing fluids38 and could be related to the stick−slip dynamics in the interfacial region when submitted to small deformations. Under the above assumptions, our model foresees a reduced dissipation of the interfacial oscillations at ultralow excitations, while the effect of viscoelasticity on the frequency of the resonance peak is much more modest. The calculated peak width (defined through the parameter Q−1) is significantly narrower in the region of the ultralow excitations of the interface, rapidly reaching a plateau at larger excitations (see Figure 3B). The qualitative agreement between theoretical and experimental results is encouraging (compare the theoretical curve of Figure 1 with the experimental results of Figure 3B). The presence of a kink in the theoretical curve of Figure 1 stems from the oversimplified nature of our model. We assumed, indeed, that below a critical threshold of the deformation, water behaves like a viscoelastic fluid, while above the threshold water shows the usual viscous behavior. A more realistic model should introduce a smooth transition between these different regimes, giving rise, in Figure 1, to a downward curvature instead of a kink. Combining eqs 26 and 27, we get a rough estimate of the elastic properties of the water interface in the anomalous 2 min region: (Q−1)max/(Q−1)min ≈ γmax ≈ (ωonλmax n /γn 1 ) . Using the experimental values of (Q−1)max (inverse quality factor at fairly large deformations) and (Q−1)min (inverse quality factor at zero deformations), we calculated the dimensionless Deborah number De ≡ ωonλmax 1 , which is a popular measure of the elastic 6446

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(6) Kim, B.; Kwon, S.; Mun, H.; An, S.; Jhe, W. Energy dissipation of nanoconfined hydration layer: Long-range hydration on the hydrophilic solid surface. Sci. Rep. 2015, 4, 6499. (7) Khan, S. H.; Matei, G.; Patil, S.; Hoffmann, P. M. Dynamic Solidification in Nanoconfined Water Films. Phys. Rev. Lett. 2010, 105, 106101. (8) Kageshima, M. Layer-resolved relaxation dynamics of confined water analyzed through subnanometer shear measurement. Europhys. Lett. 2014, 107, 66001. (9) Monaco, G.; Cunsolo, A.; Ruocco, G.; Sette, F. Viscoelastic behavior of water in the terahertz-frequency range: An inelastic X-ray scattering study. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1999, 60, 5505−5521. (10) Cunsolo, A. The THz Spectrum of Density Fluctuations of Water: The Viscoelastic Regime. Adv. Condens. Matter Phys. 2015, 2015, 137435. (11) Carpentier, S.; Rodrigues, M. S.; Costa, L.; Vitorino, M. V.; Charlaix, E.; Chevrier, J. Out of equilibrium GigaPa Young modulus of water nanobridge probed by Force Feedback Microscopy. Appl. Phys. Lett. 2015, 107, 204101. (12) Garcia, R.; Gomez, C. J.; Martinez, N. F.; Patil, S.; Dietz, C.; Magerle, R. Identification of nanoscale dissipation processes by dynamic atomic force microscopy. Phys. Rev. Lett. 2006, 97, 016103. (13) Persson, B.; Tosatti, E. Physics of sliding friction; Springer Science & Business Media: 2013; Vol. 311. (14) Scheidl, S. Mobility in a one-dimensional disorder potential. Z. Phys. B: Condens. Matter 1995, 97, 345−352. (15) Kane, J.; Ong, J.; Saraf, R. F. Chemistry, physics, and engineering of electrically percolating arrays of nanoparticles: a mini review. J. Mater. Chem. 2011, 21, 16846−16858. (16) Cheng, L.; Yip, N. K. The long time behavior of Brownian motion in tilted periodic potentials. Phys. D 2015, 297, 1−32. (17) Mallamace, F.; Corsaro, C.; Stanley, H. E. Possible relation of water structural relaxation to water anomalies. Proc. Natl. Acad. Sci. U. S. A. 2013, 110, 4899−4904. (18) Kadau, K.; Germann, T. C.; Hadjiconstantinou, N. G.; Lomdahl, P. S.; Dimonte, G.; Holian, B. L.; Alder, B. J. Nanohydrodynamics simulations: An atomistic view of the Rayleigh-Taylor instability. Proc. Natl. Acad. Sci. U. S. A. 2004, 101, 5851−5855. (19) Cottin-Bizonne, C.; Steinberger, A.; Cross, B.; Raccurt, O.; Charlaix, E. Nanohydrodynamics: The intrinsic flow boundary condition on smooth surfaces. Langmuir 2008, 24, 1165−1172. (20) Bresme, F.; Chacon, E.; Tarazona, P.; Tay, K. Intrinsic structure of hydrophobic surfaces: The oil-water interface. Phys. Rev. Lett. 2008, 101, 056102. (21) Khismatullin, D. B.; Nadim, A. Shape oscillations of a viscoelastic drop. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2001, 63, 061508. (22) Brenn, G.; Teichtmeister, S. Linear shape oscillations and polymeric time scales of viscoelastic drops. J. Fluid Mech. 2013, 733, 504−527. (23) Brenn, G. Analytical Solutions for Transport Processes; Springer: New York, 2016. (24) Joseph, D. Fluid Dynamics of Viscoelastic Liquids; Springer Verlag: New York, 1990. (25) Chhabra, R. P. Bubbles, drops, and particles in non-Newtonian fluids; CRC Press: 2006. (26) Brenn, G.; Plohl, G. The oscillating drop method for measuring the deformation retardation time of viscoelastic liquids. J. NonNewtonian Fluid Mech. 2015, 223, 88−97. (27) Miller, C.; Scriven, L. The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 1968, 32, 417−435. (28) Edwards, D. A.; Brenner, H.; Wasan, D. T.; Kraynik, A. M. Interfacial Transport Processes and Rheology. Phys. Today 1993, 46, 63. (29) Lyubimov, D. V.; Konovalov, V. V.; Lyubimova, T. P.; Egry, I. Small amplitude shape oscillations of a spherical liquid drop with surface viscosity. J. Fluid Mech. 2011, 677, 204−217.

and the strength of the critical perturbation, Bcrit, required to reach the classic viscous behavior (or the related critical oscillation amplitude Acrit). We are aware that the period of the surface oscillations and the relaxation time of the solvation cage surrounding a single water molecule occur on very different time scales. However, we are not investigating the motion of a single water molecule, rather we are considering the directional and collective motion of billions and billions of fluid particles in contact with a moving surface. The motion of this large amount of fluid is related to the statistics of continuous birth and death of short-living solvation cages that cannot be easily overcome by the extremely small applied perturbations. This fact introduces an elastic component to the collective motion of the fluid. Our differential interferometric technique seems particularly suitable for this kind of studies, since it explores the region of giant collective motions, such as the long-wavelength capillary waves, with sub-nanometer resolution of the oscillation amplitudes. Such figures are typical of spectroscopic techniques, which, on the contrary, can only investigate local properties of a system (e.g., hydrogen bonding). Finally, it is worth mentioning that along years of experiments with bubbles and drops in water we have always found that contaminants affect the interface by increasing energy dissipation and not the reverse.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b00895. Derivation of the equation of motion for an oscillating bubble (drop); some limiting expressions to calculate the real and imaginary components of the frequency of an oscillating bubble; description of the procedure used to derive the constitutive eq 25 for a nonlinear viscoelastic medium (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Antonio Raudino: 0000-0003-4827-9373 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Work partially supported by Ricerca d’Ateneo (FIR, University of Catania) funding. REFERENCES

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