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arX
iv:0
903.
2580
v1 [
phys
ics.
flu-
dyn]
14
Mar
200
9
Stick-slip dynamics of an oscillated sessile drop
Irina S. Fayzrakhmanova1,2, 3 and Arthur V. Straube∗4, 3
1Department of Theoretical Physics, Perm State University, Bukirev 15, Perm 614990, Russia2CFD Laboratory, Institute of Continuous Media Mechanics UB RAS, Korolev 1, 614013 Perm, Russia
3Department of Physics and Astronomy, University of Potsdam,Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany
4Institut fur Theoretische Physik, Technische Universitat Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany(Dated: March 14, 2009)
The dynamics of an oscillated sessile drop of incompressible liquid with the focus on the con-tact line hysteresis is under theoretical consideration. The solid substrate is subject to transverseoscillations, which are assumed small amplitude and high frequency. The dynamic boundary con-dition that involves an ambiguous dependence of the contact angle on the contact line velocity isapplied: the contact line starts to move only when the deviation of the contact angle exceeds acertain critical value. As a result, the stick-slip dynamics can be observed. The frequency responseof surface oscillations on the substrate and at the pole of the drop are analyzed. It is shown thatnovel features such as the emergence of antiresonant frequency bands and nontrivial competition ofdifferent resonances are caused by contact line hysteresis.
PACS numbers: 47.55.D-, 47.35.Pq, 47.55.np, 46.40.-f
I. INTRODUCTION
Rapid development of microtechnologies over lastdecades has manifested great interest in theoretical as-pects of contact line dynamics. The ability to predictthe motion of contact line and hence to control wet-ting processes becomes of paramount importance forapplications.1,2,3 Despite noticeable progress in under-standing the steady motion of the contact line, the un-steady motion remains significantly less explored and in-volves a number of important open questions. Of specialinterest is the role of contact angle hysteresis, which forunsteady motion may become a crucial feature in ob-taining the proper picture of the contact line motion. Inthe present study, we address this issue in the context ofoscillated sessile drop.
The dynamics of a drop on oscillated substrate hasbeen considered for many years, see recent surveys inRefs. 4,5. Recent experimental studies have shown thatthe contact line hysteresis can lead to such nontrivial ef-fects as stick-slip dynamics of the contact line6, climb-ing motion over inclined substrate,7 and motion overgradient8,9 and thermal gradient10 surfaces. These ex-perimental observations raise a natural question aboutthe role of the contact angle hysteresis, which is easy topose but rather difficult to answer. In most theoreticalstudies, the contact angle hysteresis has been either com-pletely neglected4,11,12,13 or treated in an oversimplifiedway, where the drop is similar to an oscillator with solidfriction.10,14,15,16,17 Although these solid-friction modelsreflect the qualitative picture of the stick-slip process,they do not provide satisfactory understanding of the
∗Author to whom correspondence should be addressed. Electronic
mail: [email protected]
phenomenon.To obtain deeper insight into the physics of the
stick-slip motion, a boundary condition suggested byL. M. Hocking18 can be applied. This condition, whichcaptures principal features of the contact line motion, in-volves an ambiguous dependence of the contact angle onthe contact line velocity:
∂ζ
∂t=
Λ(χ − χ0), χ > χ0,0, |χ| ≤ χ0,Λ(χ + χ0), χ < −χ0.
(1)
Here, functions ζ and χ describe the deviations of thefree surface and the contact angle from those in equilib-rium, respectively, and χ0 is the critical value definingthe contact angle hysteresis (see, e.g., Fig. 1). The fac-tor Λ, which has the dimension of velocity, characterizesinteraction between the substrate and the liquid and isreferred to as the wetting or the Hocking coefficient.
The particular case of χ0 = 0, in which the con-tact line velocity ∂ζ/∂t ∝ χ, describes no contact an-gle hysteresis.19 Different practically important situa-tions can be addressed by changing Λ. In terms of the acorresponding dimensionless parameter, for instance, λ,as in relation (6), these situations range from the com-pletely pinned contact line, λ → 0 (the contact angle canchange) to the opposite case of the fixed contact angle,λ → ∞ (the contact line is freely moving).
In the present study, we consider the dynamics ofa hemispherical drop on a normally oscillated solidsubstrate. We apply condition (1) without compromiseand focus on the role of the contact line hysteresis, whichhas been recently measured in a similar setup.6 Basedon this approach, we are not only able to quantitativelydescribe the stick-slip process but particularly to reveala new interesting feature in the contact angle evolution.This finding is very much reminiscent of the experimen-tal observations, which cannot be explained in terms
2
ambient
substrate
drop
PSfrag repla ements # Rz (#; t)FIG. 1: Problem geometry. A hemispherical drop on thetransversally oscillated substrate.
of the previously suggested theoretical models. Thepaper is outlined as follows. We start with the problemstatement in Sec. II. Section III provides the descriptionof the method we use to treat the problem. The obtainedresults are discussed in Sec. IV and summarized in Sec. V.
II. PROBLEM STATEMENT
Consider a sessile drop of incompressible liquid of den-sity ρ and kinematic viscosity ν, see Fig. 1. We are in-terested in the situation of a gaseous ambient, where itsdensity is much smaller than that of the drop and there-fore can be neglected. We assume that the solid substrateis subject to transverse oscillations with an amplitude aand a frequency ω. We admit that the drop is enoughsmall so that its shape is hardly distorted by gravity.This assumption ensures that the equilibrium drop sur-face is hemispherical to very high accuracy, with radiusR, and the equilibrium contact angle equals π/2.
The amplitude of external driving is considered smallin the sense ǫ ≡ a/R ≪ 1 and the frequency of the sub-strate oscillations is high enough: ωR2 ≫ ν. At suchfrequencies, viscous boundary layers, which arise nearthe rigid plate and near the free surface, become verythin. In other words, the frequency restriction allows usto neglect viscous dissipation in the liquid, which ensuresthat the approximation of inviscid liquid is justified.20 Onthe other hand, the frequency ω is assumed comparablewith the eigenfrequencies ωn of shape oscillations for aspherical drop of radius R. For our consideration, onlyeven eigenfrequencies are of interest, which spectrum isdefined by the relation ω2
n = 2n(2n− 1)(2n +2)σ/(ρR3),where σ is the surface tension.
Because of symmetry, we use the spherical referenceframe with the coordinates r, ϑ, α and the origin at thecenter of drop and restrict our analysis by the axisymmet-ric problem. In the accepted approximations, the fluidmotion is irrotational, which makes it convenient to intro-duce the velocity potential ϕ. As a result, the dynamicsof the liquid is described by the Bernoulli equation andthe incompressibility condition. Let r = R + ζ(ϑ, t) bethe instantaneous locus of the distorted free surface, seeFig. 1. To make a comparison with the nonhysteresis
study4 simpler, we measure the time t, length r, velocitypotential ϕ, the deviation of the pressure field p from itsequilibrium value and the surface deviation ζ in the scalesof
√
ρR3/σ, R, a√
σ/ρR, aσ/R2, and a, respectively. Asa result, the dimensionless boundary value problem isdefined by (intermediate steps can be found in Ref. 4)
p = −∂ϕ
∂t− Ω2z cosΩt, ∇2ϕ = 0, (2)
ϑ =π
2:
∂ϕ
∂ϑ= 0, (3)
r = 1 :∂ϕ
∂r=
∂ζ
∂t, p + (∇2
ϑ + 2)ζ = 0, (4)
r = 1, ϑ =π
2:
∂ζ
∂t= −
λ(γ − γ0), γ > γ0,0, |γ| ≤ γ0,λ(γ + γ0), γ < −γ0.
(5)
Here, the differential operator
∇2ϑ =
1
sin ϑ
∂
∂ϑ
(
sin ϑ∂ζ
∂ϑ
)
and γ = −(∂ζ/∂ϑ)|ϑ=π/2 is the dimensionless deviationof the contact angle from its equilibrium value, which forthe sake of brevity will be called the contact angle.
Boundary condition (3) ensures impermeability of thesubstrate for the liquid. The kinematic and the dynamicsconditions at the free surface are presented by Eq. (4).In contrast to the previous work,4 we impose a more gen-eral Hocking condition at the line of contact of the threephases, as given by Eq. (5). Thus, boundary value prob-lem (2)-(5) is similar to the previous study, except forthis hysteretic condition.
The problem involves three dimensionless parameters
Ω2 =ρω2R3
σ, λ = χ
√
ρR
σ, γ0 =
χ0R
a, (6)
which have the meaning of the squared external fre-quency rescaled with respect to the eigenfrequencies ωn,the wetting (or Hocking) parameter, and the criticalvalue of contact angle, respectively.
As the frequencies ω and ωn have been assumed com-parable, the parameter Ω is finite. A similar argumentrefers to the parameter γ0. We focus on the case of wellpolished substrate, which implies that the threshold valueχ0 is small. Being the ratio of two small parameters, χ0
and ǫ, the parameter γ0 is treated as finite. This observa-tion indicates that the contact-line hysteresis is expectedto be non-negligible even for small-amplitude oscillations.
We emphasize that condition (5) admits a number ofparticular cases. In the limiting case of perfectly polishedsurface, γ0 → 0, the boundary condition (5) is reducedto its simplified modification.4 Particularly, the contactangle remains fixed as λ → ∞, whereas the opposite case,λ → 0, describes the contact line pinned. We note that asin the latter case, the same dynamics refers to the limitof large values of γ0.
3
Finally, we stress the nontriviality of our considera-tion. Although the amplitude of oscillations is consideredsmall, ǫ ≪ 1, which allows us to linearize the govern-ing equations and simplify the boundary conditions, theoverall problem is nonlinear. The nonlinearity is ensuredby posing the Hocking condition (5) and comes into playthrough the parameter γ0 ∼ a−1, therefore the solutionis eventually amplitude dependent.
III. METHOD OF SOLUTION
To treat the formulated problem, we note that the ve-locity potential satisfies Laplace’s equation (2) and there-fore can be presented as a series in the Legendre polyno-mials. In view of impermeability condition (3), only eventerms are nonvaninshing in this expansion. We retain theterms regular at the origin and present the solutions forthe velocity potential and the consistent solutions for thesurface deviation and the pressure in the form
ζ =
∞∑
n=1
Cn(t)P2n(θ), (7)
ϕ = ϕ0(t) +
∞∑
n=1
Cn(t)P2n(θ)r2n
2n, (8)
p = p0(t) −
∞∑
n=1
Cn(t)P2n (θ) r2n
2n− Ω2z cosΩt, (9)
where θ ≡ cosϑ. Note that the summation for ζ startsfrom n = 1. The term with n = 0 is set to zero to ensurethat the drop volume is conserved. The zeroth harmonicsϕ0(t) and p0(t) = −(∂ϕ0/∂t) are nonvanishing spatiallyindependent functions, which describe spatially uniformpulsations of the velocity potential and the pressure, re-spectively. The function p0 is determined by the require-ment of conservation of the drop volume, the term ϕ0 isunimportant for the further analysis.
To distinguish between the time intervals of the contactline in motion from those in standstill, we next stick tothe following notation. Let t = t0 be the time momentwhen the contact line stops to move and t = t1 be theswitching time when it proceeds with the motion again.We next obtain the solutions describing the dynamicsat these time intervals separately and then show how tomatch the solutions.
A. Pinned contact line
We now consider the time intervals characterized bysmall values of the contact angle, |γ| < γ0. As it followsfrom the Hocking condition (5), the contact line remainsfixed during this phase of evolution (λ = 0) and we canuse the solution obtained before.4 The only point we haveto care about is that in our situation the contact line isfixed not necessarily at r = 1 but at a slightly differ-ent position characterized by ζ = ζf 6= 0, which is easy
to take into account. As a result, the solution for ζ is
presented as a superposition of the eigenmodes ζ(m)0 , a
particular solution caused by the external force, ζp, andthe term ∝ ζf , which corrects the contact line position,respectively:
ζ(θ, t) =
∞∑
m=1
Dm ζ(m)0 (θ)eiωm(t−t0) + ζp(θ, t)
+ζf(1 − 2θ). (10)
The form of the last term is chosen such that the overallexpression for ζ ensures the constant volume of the drop.A similar ansatz for the velocity potential reads
ϕ(r, θ, t) = ϕ0(t) +
∞∑
m=1
Dm ϕ(m)0 (r, θ)eiωm(t−t0)
+ϕp(r, θ, t). (11)
Here, Dm are the complex amplitudes of the eigenoscilla-tions to be determined as described in III C and ωm arethe eigenfrequencies for the drop with the pinned contactline. These eigenfrequencies are defined as the roots ofthe transcendental equation
f(ω, 0) = 0 (12)
with the function
f(x, θ) =
∞∑
n=1
αnΩ2nP2n(θ)
Ω2n − x2
, (13)
where
αn = −(4n + 1)P2n(0)
(2n − 1)(2n + 2), (14)
Ω2n = 2n(2n− 1)(2n + 2). (15)
The values Ωn have the meaning of the dimensionlesseigenfrequencies of even eigenmodes for the sphericaldrop oscillations.
The eigenfunctions for the problem with the fixed con-tact line (λ = 0) are known to be4
ζ(m)0 (θ) = 2
∞∑
n=1
nAmnP2n(θ) = Bmf(ωm, θ), (16a)
ϕ(m)0 (r, θ) = iωm
∞∑
n=0
AmnP2n(θ)r2n (16b)
with the coefficients
Amn =αn(2n − 1)(2n + 2)
Ω2n − ω2
m
Bm, n ≥ 0 (17)
B−2m = −
∞∑
n=1
αnΩ2nP2n(0)
(Ω2n − ω2
m)2 . (18)
4
Here, we introduce the normalization condition and pointout the orthogonality of the eigenfunctions
∫ 1
0
ϕ(m)0 (1, θ)ζ
(k)0 (θ)dθ = iωmδmk (19)
with δ = 1 for m = k and δ = 0 otherwise.For the problem of forced oscillations the solutions can
be expressed as
ζp(θ, t) = Re[
ζp(θ)eiΩt
]
, (20a)
ϕp(r, θ, t) = Re[
ϕp(r, θ)eiΩt
]
, (20b)
with the complex amplitudes
ζp = Ω2
[
∞∑
n=0
EnP2n(θ)
(2n − 1)(2n + 2)+ g(θ)
]
, (21a)
ϕp = iΩ∞∑
n=0
EnP2n(θ)r2n, (21b)
where
g(θ) = Fθ −1
3[1 − θ ln (1 + θ)] , (22)
En = Ω2αn
[
1 + (2n − 1)(2n + 2)F
Ω2n − Ω2
]
, n ≥ 0, (23)
F = −1
f(Ω, 0)
∞∑
n=1
2nαnP2n(0)
Ω2n − Ω2
. (24)
We indicate that sum (13) included in relations (12),(16a) and (24) converges very slowly. From the compu-tational point of view, its evaluation can be significantlyimproved if a more suitable form is used. By taking intoaccount the expansion
θ =
∞∑
n=0
αnP2n(θ), (25)
sum (13) is presented in an alternative way
f(x, θ) = θ −1
2+ x2
∞∑
n=1
αnP2n(θ)
Ω2n − x2
, (26)
which compared with the original representation (13)provides much faster convergence. Note that a similarprocedure can be applied to the sum in relation (18).
B. Moving contact line
We next deal with the time intervals of supercriticalvalues of the contact angle, |γ| > γ0, when the contactline is no longer fixed. We might build the solution in theform of series as we did in Sec. III A. This way is how-ever not worth implementing because the corresponding
eigenvalue problem is not hermitian and hence no or-thogonality condition as in (19) exists. As a result, thisapproach becomes computationally inefficient and pro-vides no advantages any more. What we do instead, isaddressing Eq. (2) for the pressure, taken at r = 1, whichallows us to figure out how the contact line evolves. Byusing the dynamic boundary condition, Eq. (4), we ex-clude the pressure and apply ansatz (7). As a result, weobtain the inhomogeneous Legendre equation
∂
∂θ
[
(
1 − θ2) ∂ζ
∂θ
]
+ 2ζ = −p0(t) + Ω2θ cosΩt
+
∞∑
n=1
Cn(t)P2n(θ)
2n. (27)
The solution of this equation is given by
ζ(θ, t) = −Ω2 cosΩt∞∑
n=1
αnP2n(θ)
(2n − 1)(2n + 2)
−
∞∑
n=1
Cn(t)P2n(θ)
2n(2n − 1)(2n + 2)+ γ
(
θ −1
2
)
,(28)
where the term ∝ γ is the general solution of the ho-mogeneous equation and the first two terms present apartial solution of the inhomogeneous equation. We notethat the integration “constant,” γ = γ(t), satisfies thedefinition of the contact angle
∂ζ
∂θ
∣
∣
∣
∣
θ=0
≡ γ. (29)
For this reason, the time-dependent function γ(t) is re-ferred to as the contact angle.
By making comparison of expressions (7) and (28) andusing relations (15) and (25), we derive a set of ordinarydifferential equations for the expansion coefficients Cn
Cn + Ω2nCn = Ω2
nαnγ − 2nΩ2αn cosΩt, (30)
which are coupled to each other through γ(t). As thefunction γ(t) is unknown, an additional relation is re-quired to close the system. The formulation of the prob-lem is completed by rewriting the Hocking condition (5),which yields
γ(t) =
S(t) + γ0, γ > γ0,S(t) − γ0, γ < −γ0,
(31)
with the auxiliary function
S(t) =1
λ
∞∑
n=1
Cn(t)P2n(0). (32)
Thus, at the stage of evolution with supercritical con-tact angles, |γ| > γ0, we numerically solve the system ofordinary inhomogeneous differential equations (30) to-gether with algebraic coupling relation (31).
5
C. Matching the solutions
To obtain the solution of the full problem, we haveto match the solutions obtained in Secs. III A and III B.The regime with the motionless contact line, which ischaracterized by subcritical values of the contact angle,|γ| < γ0, is described by expressions (10) and (11) withthe unknown complex-valued coefficients Dm. At super-critical values of the contact angle, |γ| > γ0, the con-tact line keeps moving. At this time interval, we treatEqs. (30)-(32) numerically, in terms of functions Cn(t)
and Cn(t). Next we provide relations between the coef-
ficients Dm and Cn, Cn valid at t = t0 and t = t1, atwhich the regimes are switched. At these moments, thecontact angle reaches its critical value, γ = γ0, and thetwo solutions coincide.
Consider first the moment t = t0, when the switchoverfrom the regime with the moving contact line to the onewith the pinned contact line occurs. Given the valuesCn(t0) and Cn(t0) obtained at the previous phase of mo-tion, we have to determine coefficients Dm. We multiply
Eqs. (10) and (11) by ϕ(m)0 (1, θ) and ζ
(m)0 (θ), respectively,
take into consideration integral condition (19), and ob-
tain the real, D(r)m , and imaginary, D
(i)m , parts of Dm
D(r)m =
∞∑
n=1
Amn
4n + 1(Cn − 2nEn cosΩt0) −
ζfBm
ω2m
,
D(i)m = −
1
ωm
∞∑
n=1
Amn
4n + 1
(
Cn + 2nΩEn sin Ωt0
)
.
Here, the values ωm are determined as the roots ofEq. (12) and the coefficients Amn, Bm, and En are givenby relations (17), (18), and (23), respectively.
This transformation identifies solutions (10) and (11)uniquely and allows us to determine the moment of thebackward switchover, t = t1, when the contact line startsto move again. To find out this moment, we numericallysolve the algebraic equation with respect to t1
γ(t1) − γ0 = 0. (33)
To evaluate γ we use Eq. (29) with the solution for ζ,Eq. (10), where the sum in expression (16a) should betaken in the form (26). As a result, we obtain
γ(t) = Ω2F cosΩt − 2ζf +
∞∑
m=1
Bmd(r)m (t). (34)
Here, we have introduced a complex valued function oftime dm(t) ≡ Dm exp[iωm(t − t0)] with the real and
imaginary parts denoted as d(r)m and d
(i)m , respectively.
Thus, having obtained the value t1, we are ready toproceed to the next situation.
We now turn to the consideration of the moment t = t1,when the motionless contact line starts to move. Beforewe treat Eqs. (30)-(32), we need to evaluate initial values
Cn(t1) and Cn(t1). We equate solutions (10) and (11) tothose in relations (7) and (8), which are taken at r = 1and t = t1. All the terms are presented as series in theLegendre polynomials. This can be done with the aid ofexpressions (16), (20), and (21). The coefficients on theleft and right hand sides must be equal, which yields
Cn = 2n
[
∞∑
m=1
d(r)m (t1)Amn + En cosΩt1
]
− 2ζfαn,
Cn = −2n
[
∞∑
m=1
ωmd(i)m (t1)Amn + ΩEn sinΩt1
]
.
Thus, we start from these initial values and solveEqs. (30)-(32) until the condition |γ| < γ0 is fulfilled.Suppose that γ crosses the critical value γ0 between thetime steps k and (k + 1). We need to find the momentt = t0 with the accuracy consistent with the integrationscheme and evaluate the values Cn(t0) and Cn(t0). Toestimate t0 we implement an elegant method suggestedby Henon.21 The idea is to introduce a characteristicsthat changes its sign while γ is crossing the value γ0. Asuitable quantity satisfying this requirement can be thefunction S(t), see relation (32), which turns to zero atγ(t0) = γ0. Thus, t0 as well as the values Cn(t0) and
Cn(t0) are obtained by making one additional corrective
integration step S(tk) from the values Cn(tk) and Cn(tk),which are available at time step k.
To have this idea implemented, we proceed fromEqs. (30) to the system of differential equations with thenew independent variable S and the time t = t(S) as anadditional variable
dCn
dS=
Cn
H, (35a)
dCn
dS=
Ω2n (αnγ − Cn) − 2nΩ2αn cosΩt
H, (35b)
dt
dS=
1
H. (35c)
Here, we have introduced the Henon function
H =dS
dt=
1
λ
∞∑
n=1
CnP2n(0), (36)
where Cn can be expressed from Eqs. (30).Finally, we integrate Eqs. (35) along with relation (31)
until S changes its sign. We note that while making theregular integration steps one sets H = 1. The correc-tive integration with the step S(tk) is made with H inthe form (36), which after the correction corresponds toS = 0 or equivalently to t = t1 and hence provides therequired Cn(t1) and Cn(t1).
Thus, we started with the consideration of the momentt = t0, provided the way of proceeding to the momentt = t1 and then to the next moment t = t0. To this end,we have obtained the solution over half of the period and
6
the described matching procedure can be successively re-peated to obtain the solution at longer times.
In numerical calculations, infinite number of eigen-modes in Eqs. (10) and (11) was truncated to retain Mterms. The presented results were calculated for M = 10.The control computations with the number of the eigen-modes with up to M = 20 have indicated no change inthe results. However, a further increase of M leads tothe emergence of unphysical oscillations. The numberN of Legendre harmonics retained in Eqs. (7)-(9) andEqs. (16) and (21) was chosen to be 100 in most of calcu-lations. In order to check the accuracy of calculations weperformed a number of tests with N = 150 and N = 200,which gave very close results.
IV. RESULTS AND DISCUSSION
We start our discussion by recalling the fact19 that de-spite the neglected mechanism of viscous dissipation, theHocking condition itself is dissipative. Exceptions arethe particular case of the pinned contact line (λ → 0)and the contact line freely moving (λ → ∞) over theperfectly polished (γ0 → 0) substrate. Because the sys-tem under consideration is generally dissipative, any ini-tial state approaches the terminal oscillatory state aftera certain transient. In other words, any phase trajectoryis landing at a limit cycle. Because in the case γ0 → 0and λ = O(1) the decay rate is comparable with thefrequency of oscillation,4 the transient time is estimatedto be a few periods of oscillations. For these reasons, weare mostly interested in the properties of the steady-stateoscillations.
To get an impression about the dynamics, we lookat the steady-state oscillations of the contact angle, seeFig. 2(a). When one sees the evolution of γ, it might bethought of simple linear oscillations. However, despite arelatively simple form of the observed signal, the oscilla-tions are nonlinear, which is well seen from the Fourierspectrum of the signal, Fig. 2(b). We note that althoughthe driving frequency dominates, a few higher harmonicsare nonvanishing. Another feature one can readily noticeis the absence of the even harmonics, which reflects thefact that the response is presented by an antisymmetricfunction. This antisymmetry of the steady-state oscilla-tions can be seen directly from our mathematical model.Indeed, in the terminal state the system is invariant withrespect to the transformation
t → t +π
Ω, ζ → −ζ, ϕ → −ϕ, (37)
which is easily understood if one takes into accounttwo circumstances. First, the governing equations andboundary conditions, Eqs. (2)-(5), are linear, if consid-ered at intervals with the fixed and moving contact lineseparately. Only the periodic switching between theseregimes makes the problem nonlinear. We also note thatthe problem, Eqs. (2)-(5), is inhomogeneous. However,
-3
-2
-1
0
1
2
3
46 47 48 49 50
(a)
10-3
10-2
10-1
100
101
102
0 2 4 6
(b)PSfrag repla ementst != p0 , p, 0 P (!)
FIG. 2: Characteristics of the steady-state oscillations at λ =1, γ0 = 1, and Ω = 3. Panel (a): Evolutions of the contactangle, γ(t), deviations of the free surface at the pole, ζp(t),and at the contact line, ζ0(t). The horizontal dashed lines arethe lines γ = ±γ0. The filled areas display the time intervalsof the subcritical contact angles, |γ| < γ0. Panel (b): TheFourier power spectrum evaluated for γ(t).
the inhomogeneity ∝ cosΩt changes its sign under timetransformation (37), as required. Second, we take thesame threshold value γ0 used for the advancing and re-ceding motion of the contact line. Our numerical testswith γ0 and γ1 6= γ0 for the thresholds of the advancingand receding motion, respectively, have confirmed thisstatement. For the distinct threshold values, we detectnonvanishing even harmonics, which contributions to thepower spectrum become more pronounced as γ0 and γ1
become more distinct. In the opposite case of γ1 → γ0,the even harmonics die out and we come back to theperfect antisymmetry, as in relation (37).
Along with the contact angle, we measure the devi-ations of the free surface from its equilibrium positionat the pole of the hemisphere, ζp(t) ≡ ζ(θ = 1, t) andon the substrate ζ0(t) ≡ ζ(θ = 0, t), Fig. 2(a). Thelatter characteristics shows the dynamics of the contactline. We see that the evolution of the system consistsof two interchanging regimes. During the time intervalscharacterized by supercritical values of the contact angle,|γ(t)| > γ0, the contact line keeps sliding over the sub-strate. This motion takes place until γ enters the sub-critical domain, −γ0 < γ(t) < γ0, where the contact linebecomes “frozen.” In Fig. 2(a), the time intervals of thecontact line being fixed are presented as the gray-filled ar-eas. As we clearly see, ζ0(t) = const here, whereas othercharacteristics are changing. The contact line remainsfixed until the contact angle is outside the subcriticaldomain. After that, the contact line proceeds to moveagain, etc. Thus, the contact line dynamics correspondsto the periodic sliding interrupted by the intervals of be-ing completely frozen, or, in other words, to stick-slipmotion.
We now keep the value of the wetting parameter fixed,λ = 1, and analyze how the amplitudes of ζp and ζ0
change while varying the external frequency Ω and thecritical contact angle, γ0. The corresponding dependen-cies are depicted in Fig. 3. As we see from the form ofthese response characteristics, the system demonstrateswell pronounced resonances. Because the contact line
7
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0 4 8 12 16 20 24
(a)
0
0.5
1
1.5
2
0 4 8 12 16 20 24
(b)PSfrag repla ements
p 0 0 = 0 0 = 1 0 = 3 0 = 6
= 0
FIG. 3: Amplitude-frequency response at λ = 1 and differentvalues of γ0. Panel (a): Surface deviation at the pole, ζp.Panel (b): Surface deviation on the substrate, ζ0.
motion with a finite value of the wetting parameter λ isdissipative, the resonant amplitudes remain bounded. InFig. 3(a) we also present the nondissipative limiting caseof λ = 0.
We next pay attention to the dependence on γ0. Thepartial case of γ0 = 0 corresponds to no hysteresis so thatat λ = 1 the contact line keeps moving all the time. Withthe increase of γ0, the part of period with the movingcontact line becomes less and is gradually replaced withthe regime with the fixed contact line. At large valuesof γ0, the contact line remains fixed for the most partof period, which becomes equivalent to the case of smallλ. In other words, the dynamics with the fixed contactline dominates. At λ = 0, the contact line is pinned,ζ0 = 0, and the oscillations are no longer dissipative,which results in the divergence of resonant amplitudes ofζp. We note that the case of λ = 1 is characterized by theresonant frequencies close to ωm for all γ0, where ωm arethe eigenfrequencies at λ = 0, see Eqs. (12) and (13). Asa consequence, the amplitudes of oscillations at the pole,ζp, are significantly higher than those on the contact line,ζ0.
Let us now point out another generic feature causedby the contact line hysteresis. We start with the limit-ing case of no hysteresis, γ0 = 0. We recall that in thiscase,4 the contact line remains fixed, ζ0 = 0, at certainvalues of the driving frequency, Ω = Ωar, and any valueof the wetting parameter, λ. For this reason, the val-ues Ωar are referred to as antiresonant. Such frequenciesare well recognized in Fig. 3(b). As becomes clear fromthe figure, the contact line hysteresis, γ0 6= 0, transformsthe discrete number of antiresonant points into antireso-nant bands of finite width, Fig. 3(b). With the growth ofγ0, the islands of stick-slip dynamics become narrower,whereas the regions of behavior with the completely fixedcontact line widen.
We emphasize that the width of the antiresonant bands
0
3
6
9
12
0 4 8 12 16 20 24
fixedstick-slipPSfrag repla ements 0
FIG. 4: The diagram of contact line motion on the plane (Ω,γ0). The solid lines are defined by the condition Γ(Ω) = γ0
and separate the domains of oscillations with the fixed contactline (Γ < γ0, in gray) and with the contact line moving in thestick-slip regime (Γ > γ0). Courtesy of S. Shklyaev.
is determined solely by the value of γ0 and is indepen-dent of λ, which is explained as follows. The dynamics atfrequencies within the antiresonant band corresponds tothe oscillations with the fixed contact line, as if λ = 0. Itis clear from Eq. (5), that here we have Γ(Ω) < γ0 withΓ = max γ(t), whereas outside the domain of antireso-nant behavior the opposite equality holds, Γ(Ω) > γ0.Hence, the border between the domains of stick-slip dy-namics and behavior with the fixed contact line is de-fined by the equality Γ(Ω) = γ0 or even much simpler:Γ0(Ω) = γ0, where Γ0 = Γ|λ=0. As we see, the ques-tion about the width of the antiresonant bands can beefficiently answered within the nonhysteretic model.4 Acorresponding diagram is shown in Fig. 4.
The next question to answer is if one might expectany significant difference in the dynamics for other valuesof the wetting parameter. The case of small values ofλ is out of interest because the dynamics becomes verysimilar to the case of the pinned contact line, λ → 0. Ournumerical tests show that this limit is practically reachedat λ = 1/2. As a result, the variation of γ0 demonstratesalmost no significant changes and hence the case of λ < 1brings basically nothing new.
Much more promising is the opposite situation, λ ≫1. In the nonhysteretic (γ0 6= 0) case, the contact lineis not fixed and the sliding motion is predominant. Incontrast to the case of λ ≪ 1, interaction of the dropwith the substrate is weakened. The eigenfrequenciesbecome close to the eigenfrequencies of the even modesfor a spherical drop, Ωn. In the case of hysteresis, γ0 6= 0,the system is switched between two weakly dissipativekinds of oscillations. As we have seen for the case ofλ = 1, the stage of evolution with the fixed contact lineis characterized by the resonant frequencies ωm. For thestage of sliding contact line, the resonances are foundat Ωn < ωn. Thus, a competition of the qualitativelydifferent resonances is expected for the stick-slip motion.
Our computations indicate that the described scenariowith λ ≫ 1 can be observed already at λ = 5. The cor-responding response characteristics ζp and ζ0 are shownin Fig. 5, where the competition of pairs of neighboringresonances for ζp is well seen. As in the case of λ = 1, the
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0 4 8 12 16 20 24
(a)
0
1
2
3
4
0 4 8 12 16 20 24
(b)PSfrag repla ements
p 0 0 = 0 0 = 1 0 = 3 0 = 6
= 0
FIG. 5: Amplitude-frequency response at λ = 5 and differentvalues of γ0. Panel (a): Surface deviation at the pole, ζp.Panel (b): Surface deviation on the substrate, ζ0.
growth of γ0 demonstrates convergence to the resonantcurve corresponding to the fixed contact line, λ = 0. Thetransition is however nontrivial. In contrast to Fig. 3(a),the curve associated with γ0 = 0 has resonant peaks atΩ = Ωn. As the parameter γ0 is increased, the peaks donot simply shift from Ω = Ωn toward the values Ω = ωn.This transition is accompanied by the emergence of in-termediate local maxima. Another distinction is thatthe amplitudes of resonant peaks change now nonmono-tonically with the increase of γ0. For instance, considerthe amplitude of ζp in a vicinity of the first resonance,Ω ∈ (2, 6). At γ0 = 0, we have the maximum valueζp ≈ 7.30. With the growth of γ0, the resonant ampli-tude starts to decrease and approaches its minimal valueζp ≈ 3.09 at γ0 = 1.45. The further increase of γ0 leadsto the growth and then divergence of the resonant am-plitude, as in the case of λ = 0.
The dependence of the amplitude ζ0 on γ0 and Ω isqualitatively the same as described for λ = 1 and isin agreement with the diagram of contact line motion,Fig. 4. The amplitudes of ζ0 have, however, highervalues because of weaker dissipation than those at λ = 1.
We next examine the evolution of γ and its Fourierspectrum evaluated at λ = 5 and γ0 = 3, see Fig. 6. Aswe see, the dependence γ(t) looks not only more com-plicated, but qualitatively different in comparison withthat given in Fig. 2. To avoid any confusion, we here-after stick to the following convention. We focus on halfthe period of the signal γ(t) such that γ > 0. Note thatin the case discussed in Fig. 2(a) we see a single maxi-mum. If we now go back to Fig. 6(a) we detect the birthof the second local maximum, the origin of which is dis-cussed in a few lines. As a result, the power spectrumbecomes wider than in Fig. 2(b) and the contribution ofhigher harmonics is stronger. It is important to indicatethat a very similar feature has been recently observed
-8
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4
8
19.4 19.8 20.2 20.6
(a)
10-3
10-2
10-1
100
101
102
0 4 8 12
(b)PSfrag repla ements t != P (!)
FIG. 6: Characteristics of the steady-state oscillations atλ = 5, γ0 = 3, and Ω = 9.8. Panel (a): Evolution of thecontact angle, γ(t). The horizontal dashed lines are the linesγ = ±γ0. The filled areas display the time intervals of thesubcritical contact angles, |γ| < γ0. Panel (b): The Fourierpower spectrum evaluated for γ(t).
experimentally, see Fig. 11, Ref. 6. Along with the ex-perimental study, the authors have suggested a simpletheoretical model. Although their model is able to qual-itatively explain the existence of the stick-slip motion, itfails to reproduce the two-maxima feature in the evolu-tion of the contact angle, γ(t). Although the consideredproblems are not exactly the same, the advantage of ourapproach comes into play. Our model allows us not onlyto describe the stick-slip motion itself but also to capturethe subtle feature of non-single maximum in the evolu-tion of contact angle.
To get a deeper insight into the two maxima phe-nomenon, we provide Fig. 7 evaluated at λ = 5, γ0 = 3,and Ω = 11.4. We now consider half the period of ζp(t)with ζp > 0. Figure 7(b) additionally presents the pro-files of the drop at different times as indicated with circlesin Fig. 7(a).
As we see in Fig. 7(a), the dependence ζp(t) possessestwo local maxima. One maximum, which is similarlypresent for ζp(t) in Fig. 2, concerns the stage of oscilla-tions with the fixed contact line. The second maximumis new, it relates to the stage of sliding contact line. Notethat each stage of motion is characterized by its own res-onance, which are competing as we discussed for the caseof λ = 5, Fig. 5. At γ0 = 0, the motion with the fixedcontact line is characterized by the resonances at the fre-quencies ωm, whereas for the slip motion the resonancesat Ωn become important. At Ω = 11.4 and γ0 = 0, theclosest eigenfrequencies are Ω2 = 8.49 (fixed contact an-gle) and ω2 = 10.6 (fixed contact line). If we look howthese resonant values change as γ0 is increased, we findthat at γ0 = 3 the resonances take place at Ω2 ≈ 9.10 andω2 ≈ 12.0 and the value Ω = 11.4 is well in between andclose to both of them. This explanation may also revealthe reason behind the two maxima as in Fig. 6. Despiteboth those maxima are found during the slip motion,they are caused by the competing resonances of differentnature, as we described.
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6
20 20.4 20.8 21.2
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0
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0.9
1.2
0 0.3 0.6 0.9 1.2
1
2
3
4
(b)
1234
PSfrag repla ementst x p; zp
FIG. 7: Characteristics of the steady-state oscillations at λ =5, γ0 = 3, and Ω = 11.4. Panel (a): Evolutions of the freesurface deviation at the pole, ζp(t) and the contact angle,γ(t). The horizontal dashed lines are the lines γ = ±γ0. Thefilled areas display the time intervals of the subcritical contactangles, |γ| < γ0. Panel (b): Profiles of the free surface shownat the consecutive moments of time as indicated by points 1,2, 3, and 4 in panel (a).
V. CONCLUSIONS
We have considered the dynamics of an oscillated ses-sile drop of incompressible liquid and focused on the con-tact line hysteresis. The solid substrate is subject totransverse oscillations, which are assumed small ampli-tude and high frequency. We admit that the drop isso small that its shape is not distorted by gravity andhence the equilibrium drop surface is hemispherical andthe equilibrium contact angle equals π/2. To take intoconsideration the contact line hysteresis, the boundarycondition suggested by L. M. Hocking is applied, seeEq. (1). This boundary condition involves an ambigu-ous dependence of the contact angle on the contact linevelocity. More precisely, the contact line starts to moveonly when the deviation of the contact angle exceeds acertain critical value. As a result, the stick-slip dynam-ics can be observed: the system is periodically switchedbetween the states with the sliding and the completelyfixed contact line.
The solution of the boundary value problem is pre-sented as series in the Legendre polynomials. Techni-cally, the problem is treated by building two separatesolutions valid at subcritical and supercritical values ofthe contact angle. These solutions are different and cor-respond to oscillations with the completely fixed and themoving contact line, respectively. For the fixed contactline, the problem admits an analytical solution obtainedearlier.4 In the situation with the moving contact line, aset of ordinary differential equations is obtained for ex-pansion coefficients, which are integrated numerically. Atthe critical values of the contact angle, the matching ofthe different solutions is performed. This procedure al-lows one to obtain the solution of the formulated problemat any moment of time.
Because of dissipative nature of the Hocking condition,the regime with steady nonlinear oscillations is reached.We have measured the deviations of the free surface on
the substrate and analyzed the frequency response at dif-ferent values of the wetting parameter, λ, and the criticalcontact angle, γ0. It is known that in the nonhystereticlimit, γ0 = 0, no contact line motion exists at certain fre-quencies Ω = Ωar, which are independent of the wettingparameter. For this reason, the values Ωar are referred toas antiresonant frequencies. We have shown that the con-tact line hysteresis, when γ0 6= 0, transforms this discretenumber of Ωar into antiresonant frequency bands of finitewidth. With the growth of γ0, the parameter domainsof the stick-slip dynamics become narrower, whereas theone with the completely fixed contact line grows.
We have analyzed similar frequency response for thedeviation of the free surface at the pole of the drop. Here,at relatively small values of the wetting parameter, λ, res-onant amplification of oscillations is found at frequenciesΩ ≃ ωn for all γ0, where ωn are the eigenfrequencies ofthe problem with the pinned contact line, λ = 0.
At higher values of λ, the interaction with the sub-strate is weakened. In the case of no hysteresis, γ0 = 0,the eigenfrequencies are close to the eigenfrequenciesof the even modes for a spherical drop, Ωn. We havedemonstrated that the contact line hysteresis leads toa nontrivial shift of resonant frequencies from Ω ≃ Ωn
to ωn as γ0 → ∞. For moderate values γ0 ≃ O(1),the switching between two weakly dissipative kindsof oscillations: with the sliding and the completelyfixed contact line. These stages of stick-slip motion arecharacterized by the resonant frequencies Ωn and ωm,respectively. As a result, in the interval of frequenciesΩ ∈ (Ωn, ωn) a competition of the two resonances occursand nontrivial effects can be found. Particularly, theevolution of contact angle has displayed the emergenceof an additional local maximum at half a period, whichis reminiscent of recent experimental observations,6 seeFig. 6(a).
Acknowledgments
We are grateful to S. Shklyaev for many fruitful dis-cussions, valuable comments, and providing the diagramshown in Fig. 4.
I.F. is thankful to DAAD (Russian-German MikhailLomonosov Program, project No. A/07/72463) for sup-port and to A. Pikovsky for hosting the activity. A.S.was supported by German Science Foundation, DFG SPP1164 “Nano- and microfluidics,” project 1021/1-2. Theresearch has been a part of a joint German-Russian col-laborative initiative recognized by German Science Foun-dation (DFG project No. 436 RUS113/977/0-1) andRussian Foundation for Basic Research (RFBR projectNo. 08-01-91959). The authors gratefully acknowledgethe funding organizations for support.
10
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