Disorder-induced hysteresis and nonlocality of contact line motion in chemically heterogeneous...

Disorder-induced hysteresis and nonlocality of contact line motion inchemically heterogeneous microchannels

Christophe Wylock,1, 2 Marc Pradas,2 Benoit Haut,1 Pierre Colinet,1 and Serafim Kalliadasis21)TIPs (Transfers, Interfaces, and Processes), Universite Libre de Bruxelles C. P. 165/67, Avenue F.D. Roosevelt 50,1050 Bruxelles, Belgium2)Department of Chemical Engineering, Imperial College London, London, SW7 2AZ,United Kingdom

(Dated: 16 December 2011)

We examine the motion of a liquid-air meniscus advancing into a microchannel with chemically heterogeneouswalls. We consider the case where a constant flow rate is imposed, so that the mean velocity of the interfaceis kept constant, and study the effects of the disorder properties on the apparent contact angle for eachmicrochannel surface. We focus here on a large diffusivity regime, where any possible advection effect isnot taken into account. To this end, we make use of a phase field model that enables contact line motionby diffusive interfacial fluxes and takes into account the wetting properties of the walls. We show that ina regime of sufficiently low velocities, the contact angle suffers a hysteresis behavior which is enhanced bythe disorder strength. We also show that the contact line dynamics at each surface of the microchannel maybecome largely coupled with each other when different wetting properties are applied at each wall, reflectingthat the dynamics of the interface is dominated by nonlocal effects.

I. INTRODUCTION

Wetting phenomena are of paramount significance innumerous natural phenomena and technological applica-tions. They are also central in many fields, from con-densed matter physics and fluid mechanics to materialsscience, surface chemistry and biology. As a consequence,they have received considerable attention, both theoret-ically and experimentally for several decades (see Ref. 1for a recent review).

Wetting phenomena typically involve a liquid-gas (usu-ally air) interface advancing or receding on a solid sub-strate. The wetting characteristics of a solid substrateare usually quantified in terms of the contact angle at thethree-phase conjunction, defined as the angle between theliquid-gas interface and the wetted area of the substrate.The intersection between the liquid-gas interface with thesolid substrate is commonly referred to as contact line.When one of the two fluids moves against the other, thecontact line has to move relative to the solid boundary.A problem of dynamic wetting that has attracted a lot ofattention over the years is the history-dependence of thecontact line, and in particular, the hysteresis behaviorof the contact angle. Contact-angle hysteresis is usu-ally characterized by a discontinuous jump between themeasured contact angle of an advancing and a recedingcontact line in the zero velocity limit. It is now generallyaccepted that the origin of such hysteresis is mainly dueto the presence of either chemical or topographical het-erogeneities on the substrate2–4 and it is related to thepresence of a wide range of possible metastable statesthat describe the equilibrium configuration of the con-tact line5. Several approaches for different wetting situa-tions have been considered during the last years in orderto understand such phenomenon. Examples include con-tact angle hysteresis in droplet motion induced by het-erogeneous substrates6–8, or by shear flow9. However, a

satisfactory description of the hysteresis in terms of thedisorder properties of the substrate, e.g. density of the de-fects or disorder strength, is still lacking (see for examplethe discussion on hysteresis in the recent review by Bonnet al.1). Likewise, another important issue in the physicsof wetting phenomena is whether one can consider thedynamics of the contact line as a local process, i.e., aprocess that mainly depends on the local velocity closeto the contact point, or, by contrast, a process dominatedby non-local effects, i.e., long-range correlations along theliquid-gas interface that are a consequence of the globalflow rate conditions applied at the system.

In the preset study we address these questions by us-ing as a model system for contact line motion a two-dimensional (2D) liquid-air meniscus advancing into a 2DHele-Shaw-like microchannel with chemically disorderedsurfaces (our system is assumed translationally invariantin the transverse direction). The displacement of a fluidby another one in a microchannel is a subject of increas-ing interest in many engineering applications, especiallywithin the framework of the rapidly growing fields of mi-crofluidics and miniaturization of chemical devices10. Anumber of fundamental questions has also attracted sig-nificant attention over the last few years in particularin relation to the effects of the microchannel surfacesupon the contact line dynamics of a liquid-air menis-cus advancing through the channel. These include, forexample, how the wetting and/or roughness propertiesof the solid boundaries may induce complex phenomenalike contact angle hysteresis4,7,11,12 or contact line pin-ning/depinning13–17. As we noted earlier, a satisfactorydescription of hysteresis in terms of the accessible con-trol parameters of the system, such as strength of thedisorder or distribution of defects, is still lacking. Infact, the same is true regarding pinning/depinning ef-fects which however, are beyond the scope of this study.Clearly, achieving a systematic and rational description

This is the postprint version of an article appeared in Phys. Fluids 24, 032108 (2012). It may be found at http://dx.doi.org/10.1063/1.3696860. Copyright 2012 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

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of such complex wetting phenomena in terms of the disor-der properties is crucial, both from a fundamental pointof view and engineering applications.

Our microchannel consists of two parallel plates sepa-rated by a narrow distance which is much smaller thanthe dimensions of the channel plates, and is usuallysmaller than the capillary length `c, denoting the rel-ative importance between surface tension and gravity,which for most liquids is `c ∼ 1 mm18. In this case, (a)gravity does not play a significant role, and (b) channelsurface heterogeneities and nonlocal effects arising froma competition between the relevant forces acting on theinterface, e.g. surface tension and driving force, may playa crucial role in the contact angle dynamics. Regardingnonlocality, of particular interest is the study of possiblecorrelations between the contact line dynamics at eachsurface plate, or more specifically, to understand underwhich conditions both contact lines can be considered in-dependent from each other when they are perturbed byrandom heterogeneities, or otherwise they follow a cou-pled dynamics as a consequence of nonlocal effects.

We consider here a regime where diffusion processesdominate over advection processes. This means that anytypical velocity is assumed to be sufficiently small so thatthe system is considered to be close to an equilibriumstate. To model this situation, we make use of a diffuseinterface model that enables contact line motion by dif-fusive transport of mass. The dynamics of the process isthen described in terms of the model B (if we follow theHohenberg and Halperin terminology19), and the wettingproperties are incorporated into the model via a surfaceenergy term20 that takes into account solid-fluid interac-tions. By imposing forced flow conditions, i.e., constantflow rate we focus on the effect of the disorder on thehysteresis of the contact angle and on the correlation be-tween the dynamics of the contact lines of both walls.We show that the walls’ disorder produces an importanteffect on the measured apparent contact angle, inducinga hysteresis behavior between the advancing and reced-ing motion which, in turn, is enhanced as the strengthof the noise is increased. We also show that there existsa coupled dynamics between both contact lines which isparticularly important when different wetting propertiesare applied at each surface channel. Although our studyis restricted to a 2D system, it is a first step towardsthe understanding of the influence of chemical hetero-geneities on the contact line dynamics in more complex,e.g. three-dimensional (3D) systems.

The structure of the paper is as follows. The problemformulation, physical set-up definition and the diffuse in-terface model we adopt are outlined in Sec. II. Our re-sults on contact angle hysteresis and nonlocality are pre-sented in Sec. IV. Finally, conclusions and perspectivesare given in Sec. V.

FIG. 1. (Color online) Sketch of the profile geometry for a2D meniscus in a 2D chemically heterogeneous microchannel.The bottom panel shows how the contact angle is defined inour numerical results: the red solid line represents a quadraticfit to the interface points (solid circles), which allows to ex-trapolate the contact point (star). The contact angle is thendefined as the angle between the line tangent to the interfaceat the contact point (dot-dashed line) with the wetted partof the substrate.

II. FORMULATION

A. Problem definition

Figure 1 shows a sketch of the problem definition. Weconsider a 2D liquid meniscus advancing into a 2D mi-crochannel with chemically disordered walls. We intro-duce a Cartesian coordinate system (x, z) where z is thestreamwise coordinate in the direction of the motion andx is the crosstream coordinate. The location of the liquid-air interface is denoted as h(x, t). The length and widthof the microchannel are denoted as Lz and Lx, respec-tively, with the lower–upper walls located at x = 0 andx = Lx, respectively. The inlet and the outlet of the mi-crochannel are located at z = 0 and z = Lz, respectively.Finally, the surface wetting properties of the two wallsare assumed to be randomly distributed.

B. A diffuse interface model

A Cahn-Hilliard phase-field model is used to describea system of two phases separated by a diffuse inter-face. Phase-field models have been used successfully overthe last few years in a wide variety of fluid flow prob-lems, such as solidification phenomena21, Saffman-Taylorinstabilities in Hele-Shaw flows22, or wetting phenom-ena16,23,24 (see also the review in 25). Regarding wettingphenomena in particular, one of the advantages of usingphase-field models for their description is that the diffusenature of the interface regularizes the divergent viscousdissipation associated with a moving contact line in the


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FIG. 2. (Color online) (a) Schematic representation of thephase-field profile in the z direction at a given value of x.The two limiting values ±φe represent the two phases of thesystem, and the phase field changes continuously from one tothe other in a finite region of width ε, the diffuse interface. (b)The effect of introducing a non-zero gradient of the chemicalpotential µ (via e.g. external forcing) is to destabilize one ofthe two phases in the potential double-well V (φ).

framework of the classical fluid mechanics approach26.Another way to put it is that because of the diffuse char-acter of the interface (effective) slip appears naturallythrough the introduction of the interface thickness andthere is no need to impose it27. On the other hand, work-ing with diffuse-interface models may present also somedifficulties arising from the fact that real systems haveactually a much thinner and sharper interface. For thisreason, one of the main requirements when computingphase-field models is to reach the so-called sharp inter-face limit (see for example the recent work in 28). Froma numerical point of view, this means that the resultsdo not depend on the thickness of the diffuse-interface,and therefore they are comparable to the results thatone would obtain by using a classical fluid mechanics ap-proach.

1. Equilibrium properties

In the phase-field approach, a locally conserved field,denoted as φ, plays the role of an order-parameter by

taking two equilibrium limiting values +φe and −φe thatrepresent the liquid and air phases, respectively (seeFig. 2a). The interface position, h(x, t), is then locatedat the points where the order parameter vanishes, i.e.,φ[x, h(x, t)] = 0. The equilibrium properties of the modelare based on a Ginzburg-Landau formulation, where thetotal free energy of the system is given by

F(φ) =∫

Ω

drf(φ) =∫

Ω

dr[V (φ) +

ε2

2|∇φ|2

], (1)

where f(φ) is the free energy density, Ω corresponds tothe system domain, and the potential V (φ) is usuallychosen as:

V (φ) = −12αφ2 +

14βφ4, (2)

with α and β being positive constants in such a waythat the equilibrium value for the phase field is givenby φe =

√α/β. The double-well form of the potential

combined with the square gradient term of the free en-ergy ensures the existence of a well-defined interface witha width ζ = ε/

√α. In this formulation, the chemical po-

tential is defined as µ = δF/δφ = V ′(φ)− ε2∇2φ, so thatone of the two phases can be destabilized in favor of theother, and hence forcing the interface to advance by im-posing a non-zero chemical potential difference, ∆µ 6= 0,between the inlet of the system and the interface position(see Fig. 2b). In the case of a flat interface at equilibrium(∆µ = 0), the profile of the order-parameter is given by

φ(z) = −φe tanh(

z − h√2ζ

)(3)

where z is the direction perpendicular to the interfacelocated at z = h. Also, the surface tension of the liquid-gas interface σlg can be easily obtained as the excess freeenergy per unit surface area due to the inhomogeneity inφ29:

σlg = ε2

∫dz

(dφ

dz

)2

=2√

23

ζαφ2e. (4)

To take into account the wetting properties of the walls,the free-energy equation (1) is modified by adding anextra term associated with solid-fluid interactions20

F(φ) =∫

Ω

dr[V (φ) +

ε2

2|∇φ|2

]+

∫s

dsfs(φs), (5)

where s denotes the solid surface. The function fs(φs) inthis additional term which enables us to introduce wet-ting effects into the model is related to the molecularinteractions between fluid and solid, and it is usually ex-panded as a power series in φs

26, where φs represents thevalue of the phase field at the walls. In the present studywe shall keep only the first order term30 and write:

fs = −aφs, (6)


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where a describes the preference of the wall for eitherthe liquid phase (a > 0, hydrophilic conditions) or theair phase (a < 0, hydrophobic conditions). It is impor-tant to emphasize that the linear term in the expansionof fs turns out to be sufficient to describe partial wet-ting situations29,30 as the ones we are going to considerhere. Note, however, that for situations closer to com-plete wetting (i.e. contact angles close to zero) higherorder terms might need to be included in order to avoideffects induced by the formation of mesoscopic layers onthe walls27.

We non-dimensionalize our system by choosing the fol-lowing dimensionless variables:

φ∗ = φ/φe, r∗ = r/Lx, (7a)f∗ = f/(φ2

eα), f∗s = fs/σlg. (7b)

By minimizing the corresponding dimensionless free en-ergy functional, Eq. (5), the equilibrium conditions aregiven by the relation −φ + φ3 = C2

h∇2φ, which givesthe equilibrium profile for the phase field φ, and cor-responds to setting the chemical potential to zero, i.e.,µ = δF/δφ = 0, and the wetting boundary condition20,31

nx ·∇φ|s =γ

Ch

dfs

dφs= − γ

Cha, (8)

where the asterisks have been dropped and we havedefined the Cahn number as Ch = ζ/Lx, and γ =2√

2φe/3σlg. Here, nx is the unitary outward normalvector to the wall. Note that both the condition µ = 0and Eq. (8) describe the system at equilibrium when noflux is imposed at the inlet. For a constant value of a,the corresponding static equilibrium contact angle, θe,can be obtained by making use of the Young-Dupre rela-tion, cos θe = (σsg − σsl)/σlg, where σsg and σsl are thesolid-gas and solid-liquid surface tensions, respectively,giving rise to the following relation20,29,31,32:

cos θe =12

[(1 + A)3/2 − (1−A)3/2

], (9)

where we have defined A =√

2γa. In this approach, thedifferent surface tensions σsg and σsl can be obtainedby integrating the free energy per unit area along thecorresponding interfaces, solid-gas and solid-liquid29.

2. Statistical properties of the chemical disorder

A(z) will be taken as a stochastic variable and willplay the role of a quenched disorder of different wettingproperties. Of particular interest is the relation betweenthe stochastic variable A(z) and the resulting variableθ(z) representing the local equilibrium contact angle ateach point z. To this end, we shall consider a Gaussianwhite noise for the variable A with mean Am = 〈A〉 anddelta-correlated, 〈A(z)A(z′)〉 = σ2

Aδ(z − z′), where σA is

the disorder strength (standard deviation). By assum-ing small disorder strengths, σA 1, Eq. (9) can belinearized around the mean value Am, giving rise to:

cos(θ) ' cos(θm) + κ(A−Am), (10)

where the mean contact angle θm is given by Eq. (9) byreplacing A with Am, and we have defined the constantκ as:

κ =34

[(1 + Am)1/2 + (1−Am)1/2

]. (11)

It can be readily seen that in the limit of small disorderstrength, the new variable θ follows a Gaussian distribu-tion with mean θm and standard deviation given by:

σθ 'κ

sin θmσA. (12)

Figure 3 depicts the statistical properties of the variableθ calculated numerically from Eq. (9) given a sample ofN = 2 ·106 values of A with mean Am = 0 and Am = 0.4,and for different disorder strengths ranging from σA = 0up to σA = 0.1. These numerical results are compared tothe analytical prediction given by Eq. (12) showing verygood agreement. We observe that for the case of Am =0.4, the variable θ can be considered to be Gaussian forσA . 0.1.

3. Conserved dynamics

The dynamics of the phase field is assumed to follow aconserved equation based on a time-dependent Ginzburg-Landau Hamiltonian (model B in the Hohenberg andHalperin terminology19), which in dimensional form is:

∂φ

∂t= ∇·M∇µ = ∇·M∇

(−αφ + βφ3 − ε2∇2φ

), (13)

where µ is the chemical potential defined earlier, andM(φ) is a mobility parameter that in general depends onφ. Here, we are using the so-called one-sided model22,33where the mobility is taken as M = M0 in the liquidphase (φ > 0) and M = 0 in the air phase (φ < 0). Toadvance the interface we consider a driven system, wherethe mean velocity of the interface is fixed to a constantvalue. The relevant boundary condition at the inlet isobtained by fixing the gradient of the chemical potentialat the left-hand side of the system:

nz ·∇µ|z=0 = −vm, (14)

where nz is the unitary outward normal vector at theinlet of the microchannel. By imposing this boundarycondition, the average velocity of the interface, V =1

Lx

∫x

dx∂th(x, t) takes a positive value whenever vm > 0(the liquid advances in the channel) and a negative valuewhenever vm < 0 (the liquid recedes).


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FIG. 3. (Color online) Statistical properties of the randomcontact angle θ given by Eq. (9) when A is a random vari-able with Gaussian distribution of mean Am and standarddeviation σA. Panels (a) and (b) show the probability den-sity function of the variable θ calculated by using two differentstrengths σA for Am = 0 and Am = 0.4, respectively. Squaredand circle symbols correspond to the numerical data while thedashed line corresponds to a normal distribution N(θm, σθ),with θm and σθ given by the analytical approach. The corre-sponding insets show the comparison between the numericallyobtained standard deviation σθ with the analytical expressiongiven by Eq. (12).

As before, we shall work in dimensionless variables andwe use Eq. (7) in addition of ζ2/M0α as the characteristictime scale to get the non-dimensional equation:

∂φ

∂t= Ch

2∇ ·m∇(−φ + φ3 − Ch

2∇2φ), (15)

where the non-dimensional mobility now takes the valuem = 1 in the liquid phase and m = 0 in the air phase.We also take the initial position of the interface locatedat z = zinit, so that the initial condition for the phasefield reads,

φ(t = 0) =

+1 z ≤ zinit,−1 z > zinit,

(16)

and we assume the usual no flux boundary conditions onthe walls, i.e. nx ·∇µ|s = 0, in addition to Eq. (8).

This approach enables contact line motion by diffu-sive interfacial fluxes. It should be noted that we areconsidering a large diffusivity limit, where any possibleadvection effect is not taken into account, and the sys-tem is considered to be in a close-to-equilibrium state.This is a reasonable assumption as long as the velocity ofthe interface, which we take as the typical velocity in thesystem, is sufficiently small. More precisely, if we intro-duce a Peclet number, Pe, defined as the ratio between

advection and diffusion Pe ∼ V Lx/M , we are consider-ing the regime of small Peclet numbers, Pe 1. Forhigher values of Pe, the model would have to be appro-priately modified to include an advection term coupledto the Navier-Stokes equation25,27,30,34,35, obtaining thenwhat it is also known as model H in the Hohenberg andHalperin terminology19– note that the model B we areusing here can be obtained from model H in the limit ofPe → 0 (see Appendix A for details).

III. NUMERICAL SCHEME

The dimensionless phase-field model equation (15) isintegrated numerically in a 2D domain of size `x × `z byimposing the wetting boundary conditions (8) at bothwalls, the driving condition (14) at the inlet of the mi-crochannel, the initial condition (16), and the no-fluxboundary condition at the walls. The size of the sys-tem in dimensionless variables is `x = 1 and `z = 5.We use an explicit fourth-order Runge-Kutta algorithmwith adaptive time stepsize, and a rectangular spatialdiscretization with grid ∆x = ∆z = 0.05. A smaller spa-tial grid ∆x = ∆z = 0.025 has also been used but itdoes not alter the numerical results. All numerical in-tegrations presented in this study have been performedby setting the Cahn number to Ch = 0.07 and Am = 0.4(unless otherwise specified) as the mean value for the wet-ting properties of the walls, giving rise to the equilibriumcontact angle θe ' 54. The value of the spatial noise Aat each point z is numerically imposed as the followingdiscretized white noise:

A =σA√Lx∆z

N(0, 1) + Am, (17)

where N(0, 1) denotes a Gaussian variable of zero meanand variance one. Note that the factor

√Lx and

√∆z

in the above equation originate from the variables non-dimensionalization and the spatial discretization, respec-tively.

The interface position, h(x, t), is estimated by alinear interpolation of the zero of the phase field,φ[x, h(x, t); t] = 0. This allows us to estimate the po-sition of the contact point of the liquid-air meniscus atthe wall surface by using a parabolic extrapolation of thenearest points of the meniscus. Once the location of thecontact point is determined, the apparent contact angle,θ, is defined as the angle formed by the wetted part of thewall with the tangent line to the meniscus at the contactpoint (see Fig. 1).

As we impose a constant flow rate condition, the accu-racy of our numerical scheme was monitored by examin-ing dynamically the following quantities,⟨

∂φ(x, z; t)∂t

⟩, 〈|φ(x, z)|〉, (18)

where 〈·〉 = 1/(`x`z)∫ `x

0

∫ `z

0·dxdz. Note that the quan-

tity on the left in (18) should be constant whereas the


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quantity on the right should fluctuate around a constantvalue. Both magnitudes turn out to be good indicatorsto estimate the order of the numerical error. Concerningthe discontinuous mobility function m(φ), let us remarkthat m is taken to be constant for φ ∈ [1,−0.9], so thatit is actually continuous along the interface area.

Finally, an important numerical issue that natu-rally arises in diffuse interface models is whether thesharp-interface limit can be numerically reached or not.As pointed out in Sec. II B, the achievement of thesharp-interface limit is a necessary condition for diffuse-interface numerical integrations, where the interface isusually much thicker than the actual one, and guaranteesthat the numerical results do not depend on the interfacethickness. As it has been shown in the recent work byYue et al.28, a numerical convergence criterion for suchlimit can be obtained in terms of the typical diffusionlength of the process and the Cahn number Ch. Moreprecisely, these authors have shown that the sharp inter-face limit is numerically reached as long as the relationCh < 4S holds, where S = `D/Lx, with `D being the dif-fusion length. In our case we can write `D ∼

√M0/φe,

and in order to satisfy such numerical criterion, we haveset the value S = 0.05, so that we have the numericalrelation Ch = 0.07 < 4S = 0.2. We can therefore ensurethat the numerical results presented in this study do notdepend on the interface thickness. Figure 4 also showsthe numerically obtained interface profile by choosing dif-ferent values of the Cahn number ranging in [0.03, 0.11],observing good convergence as Ch → 0.

IV. RESULTS AND DISCUSSION

A typical evolution of the advancing meniscus over achemically heterogeneous channel with σA = 0.1 anddriving force vm = 0.0025 is shown in Fig. 4. As theliquid phase advances, both the apparent contact angleat the top [θt(t)] and bottom [θb(t)] walls fluctuate withtime. Our purpose in this section is to study both thedynamics and statistical properties of the apparent con-tact angles, and how their behavior depends upon thedisorder strength σA and driving force vm.

A. Nonlocal effects in the contact line dynamics

As emphasized in the Introduction, an important ques-tion regarding wetting processes in microchannels iswhether the dynamics of the interface is dominated bynonlocal effects arising from the flow rate conditions ap-plied at the system. To investigate such effects on theapparent contact angle, we start by analyzing possiblecorrelations between contact angles at both walls. To thisend we consider a homogeneous top wall and a heteroge-neous bottom wall with Am = 0.4 and different disorderstrengths, and we examine how the dynamics of the con-tact angle at the top wall is affected by the disordered

FIG. 4. (Color online) Successive snapshots of the liquid-airmeniscus invading a chemically heterogeneous microchannelwith σA = 0.1, vm = 0.0025, and Am = 0.4 (θe ' 54).The bottom panel shows the interface profile computed byusing different values of the Cahn number, ranging between[0.03, 0.11].

50 60 70 80 90 100 110 120 13056

60

64

t

θb(t)

91

91.5

92

θt(t)

59

60

61

62

θt(t)

σA = 0.05σA = 0.1

(a) Homogeneous top wall

Homogeneous top wall

(c)

(b) Am = 0

Disordered bottom wall Am = 0.4

Am = 0.4

FIG. 5. (Color online) Temporal dynamics of the apparentcontact angle of the homogeneous top wall θt(t) [(a) and (b)]and the disordered bottom wall θb(t) [(c)]. Solid and dashedlines correspond to different disorder strengths applied at thebottom wall, namely σA = 0.05 and 0.1, respectively. (a)and (b) correspond to a homogeneous wall with Am = 0 (dif-ferent from the bottom wall), and a homogeneous wall withAm = 0.4 (equal to the bottom wall), respectively. The meanvelocity is vm = 0.005 in all cases.


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bottom wall. More precisely, correlations between wallswill be quantified by looking at the correlation factor:

ρc =〈θtθb〉 − 〈θt〉〈θb〉

σtσb, (19)

where 〈··〉 denotes time average and σ2n = 〈θ2

n〉−〈θn〉2 forn = t, b.

We first consider the case of the top wall having thesame mean value for Am as the bottom wall, Am = 0.4,and we choose two different disorder strengths for thebottom wall, namely σA = 0.05 and 0.1 [see Fig. 5(a)].By using the above expression we obtain that ρc = 0.04and 0.09 for σA = 0.05 and 0.1, respectively. We there-fore conclude that in this situation both walls can beconsidered to be uncorrelated with each other, observ-ing that the top contact angle is affected only by theusual fluctuations due to the lattice discretization. Notealso that the mean contact angle value at the top wall(〈θt〉 ∼ 60) does not depend on the disorder strengthapplied at the bottom wall. This is an important pointthat will be used in the next section to obtain averagedcontact angles. We also note that similar results havebeen obtained for smaller systems (Ch = 0.11) indicatingthat in these wetting conditions, both walls can indeedbe considered as independent from each other.

Interestingly, when we repeat the same analysis ina setting having different mean wetting values betweenboth walls, i.e., we choose Am = 0 (neutral wetting) atthe top wall and Am = 0.4 at the bottom wall, we findρc = −0.54 and −0.72 for σA = 0.05 and 0.1, respec-tively. This is a clear indication that both walls are nowmuch more correlated with each other. Indeed, as it isshown in Fig. 5(b), the dynamics of the top contact angleis now coupled to the bottom one: fluctuations above themean value occurring at the bottom contact angle are re-flected as fluctuations below the mean value at the topcontact angle, and vice versa. This effect is also evidentfrom the negative sign of the correlator ρc. Note alsothat the mean value at the top wall (〈θt〉 = 91.4) doesnot depend on the strength σA, and is slightly higherthan the expected neutral value of 90. It is worth men-tioning that this effect largely depends now on the gapbetween the walls, obtaining, for instance 〈θt〉 = 92.6for Ch = 0.11.

From a physical point of view, this nonlocal effect canbe understood in terms of the mass conservation in thesystem. Consider first an ideal situation where both thetop and bottom mean contact angle are around 90 (neu-tral wetting), so that the interface position can be ex-pressed as h(x, t) ∼ V t + h0(θt, θb, x) with V being themean interface velocity and h0(θt, θb, x) the interface pro-file which depends upon both contact angles (note thath0 = 0 for θt = θb = 90). If we consider that both con-tact angles are fluctuating in time with respect to theirmean values, θt = 〈θt〉+δθt(t) and θb = 〈θb〉+δθb(t), theconstant flow rate condition implies that

∫x

dx∂th(x, t)has to remain constant in time, and we therefore have

the relation,

˙δθt = −Ib

It

˙δθb, (20)

where the dot denotes time differentiation and In =∫x

dx∂θnh0|θn=〈θn〉 for n = t, b. In more physical terms,

the above expressions suggest that any fluctuation above(below) of the mean value at the bottom contact pointwill produce a change on the local curvature of the in-terface with the corresponding decrease (increase) of thechemical potential difference across the interface. Thisin turn will induce a local decrease (increase) of the localvelocity which will need to be compensated with a fluctu-ation of opposite behavior at the top contact point. Asa result, the top contact point is moved away from itsequilibrium configuration, and a local diffusive processwill occur trying to pull the contact point back to equilib-rium. Our numerical results indicate that this is indeedthe case when the top contact angle is fixed around theneutral value and the bottom contact angle is fluctuat-ing around a different (smaller) mean value [cf. Fig. 5(b)].In the other case, however, when both mean contact an-gles are equal and much smaller than the neutral one[cf. Fig. 5(a)], the dynamics at the top contact point islargely affected by the local pulling force due to the wet-ting properties at the surface, with the eventual loss ofthe nonlocal character.

B. Disorder-induced contact angle hysteresis

We now proceed to study the dynamics of the appar-ent contact angle for both an advancing (vm > 0) andreceding (vm < 0) motion of the contact line, with bothwalls being chemically disordered with the same meanvalue, Am = 0.4. For this purpose, we shall perform adetailed statistical analysis of the time fluctuations of thecontact angle by generating a sufficiently large number ofdifferent chemical disorder realisations.

In the advancing case, the position of the interface ismonitored during a time interval T , and at each timewe calculate the apparent contact angle at both top andbottom walls, i.e., θt(t) and θb(t), respectively, and alsothe mean velocity of the interface, V =

∫x

dx∂th(x, t).We then switch the sign of vm and repeat the processfor the receding interface and during the same time in-terval as before. The resulting set of apparent contactangles and velocities are averaged over time for each caseof motion, and also averaged over 20 different disorderrealizations, obtaining 〈θt〉 and 〈θb〉 for the averaged topand bottom contact angles, respectively, and 〈V 〉 for themean interface velocity. As shown in Sec. IVA, sincewe are considering both walls with the same mean valuefor the wetting properties (Am = 0.4), the top and bot-tom contact angles can be treated as independent vari-ables, and therefore, we can also average them to get:〈θ〉 = (〈θt〉+ 〈θb〉)/2. Finally, we look at the influence of


8

FIG. 6. (Color online) (a) Bottom contact point position asfunction of time, hb(t), for an averaged velocity of 〈V 〉 = 0.003and disorder strength σA = 0.1. The red solid line shows adata fit to a linear function hb(t) = at+b. (b) Bottom contactpoint time fluctuations obtained as δhb(t) = hb(t)−hb(t). (c)Bottom contact angle as function of time.

the disorder strength on the contact angle by taking dif-ferent values of σA ∈ [0, 0.1]. It is worth to remark thatthe actual mean velocity 〈V 〉 observed after disorder av-eraging it is slightly different than the imposed numericalvalue for the driving force vm. To be more precise, due tothe miscibility gap ∆φe = 2φe = 2 (in the dimensionlessequations) we observe that 2〈V 〉 ∼ vm for all the valuesof the disorder strength.

Figure 6 shows the time evolution of the advanc-ing bottom contact point hb(t). As a consequence ofthe forced flow boundary condition we observe a lineargrowth which can be described as hb(t) = hb(t) + δhb(t),where δhb(t) represents the small fluctuations due to thechemical heterogeneities, and hb(t) represents the meangrowth of the contact point which can be approximatedby a linear data fit [cf. Fig. 6(a)]. By plotting then thecontact point fluctuations only, i.e., δhb(t) = hb(t)−hb(t),we observe that the motion of the contact point is corre-lated with the corresponding contact angle: An increaseof the local position in time (and hence increase of thelocal velocity) is reflected as a decrease on the contactangle and vice versa [see Figs. 6(a) and (b)]. As weshow below, this local dynamics at the contact point isenough to induce observable (quantitative) hysteresis atthe contact angle level. It is important to emphasize thatstrictly speaking we do not observe here clear stick-slipmotion (i.e., a situation where the contact point remainspinned for a finite period of time, and then suddenly de-pins and advances into another position). Note that anypossible pinning-depinning motion induced by the weak

FIG. 7. (Color online) (a) Averaged contact angle 〈θ〉 as func-tion of the averaged mean interface velocity 〈V 〉 for differentdisorder strengths σA. Negative or positive velocities repre-sent receding or advancing contact line motion, respectively.The dashed line corresponds to a data fit of homogeneous case(σA = 0). Typical errors for the contact angle in these resultsrange between 0.01, observed at small disorder strengths, upto 0.1, observed for the largest disorder strength. (b) Hys-teresis gap ∆θ versus disorder strength σA. The dashed linecorresponds to a linear data fit.

hydrophilic heterogeneities we are dealing with is actu-ally being smoothed out by the imposed constant flowrate. In this sense, we could observe stronger pinningeffects had we considered, for example, a combinationof hydrophilic and hydrophobic heterogeneities8 or topo-graphical defects15,16. We leave such an extension as afuture work.

The results for 〈θ〉 as a function of 〈V 〉 for both advanc-ing and receding motions and different disorder strengthsare presented in Fig. 7(a). We first note that in the ho-mogeneous case (σA = 0), the averaged contact angleapproaches the same value, 〈θ〉 ∼ 58.5, when both thelimit of 〈V 〉 → 0+ and 〈V 〉 → 0− are taken. Therefore,as expected, when the disorder in the system is switchedoff, these two limits are identical with each other andthere is no hysteresis behavior. As we increase the disor-der strength, however, these two limits no longer matchand we observe that the apparent contact angle suffers ahysteresis effect which is enhanced by the chemical disor-der. The linear behavior of the homogeneous case is splitinto two linear branches separated by a gap that dependson the disorder strength. This behavior is in qualita-tive agreement with the experimental results shown inRef. 12. To quantify such behavior, we fit the numerical


9

data of both the advancing (+) and receding (-) branchto a linear function, 〈θ〉± = m±〈V 〉± + θ±, and the hys-teresis gap is defined as:

∆θ = |θ+ − θ−|. (21)

Figure 7(b) depicts ∆θ as function of the disorderstrength σA, confirming that the hysteresis behavior isenhanced by increasing disorder strength. Note thatlarger fluctuations on the contact angle, i.e. larger dis-order strength induce a larger error on the fitted param-eters. Noteworthy is that our numerical results seem topoint out a linear behavior between the hysteresis gapand the disorder strength, which is in agreement withprevious studies on contact angle hysteresis8. In partic-ular, by fitting the numerical data to a linear function atsufficiently large disorder strengths, we find the relation∆θ = aσA + b with a = 12.2 and b = 0.16 [see dashedline in Fig. 7(b)].

Finally, it is worth mentioning that we have also per-formed simulations at higher velocities. We observed thatfor sufficiently high velocities, both in the advancing andreceding case, the different values of the contact anglescalculated for each disorder strength collapse onto thehomogeneous curve. This indicates that at high veloc-ities, the contact line cannot distinguish between weakor strong disorder, and as a consequence the averagedbehavior corresponds to the homogeneous case. This isactually a very common situation observed whenever aninterface is driven through a medium that is affected bya static (quenched) disorder (see for instance Ref. 36).At sufficiently large values of the driving force, there is aclear scale separation between the motion of the advanc-ing/receding front and the local fluctuations of the disor-der. As a consequence, the quenched disorder behaves asan effectively temporal white noise with the eventual lossof any disorder-induced complex behaviour. However, asit was emphasized in Sec. II B, the results presented hereare expected to be valid in a regime of low velocities, i.e.small Peclet numbers, where the diffusion mechanism isdominant over advection. Hence, strictly speaking for thehigher velocity regimes and description of the motion inthe region of larger Peclet numbers, an advective termshould be introduced27,34,35.

V. CONCLUSION

We have examined numerically the effects of chemi-cal heterogeneities on the forced motion of the contactlines of a liquid-gas meniscus invading a microchannel un-der forced flow (constant flow rate) conditions. We havemade use of a diffuse-interface (phase-field) model thattakes into account the wetting properties of the channelsurfaces. We examined how the contact angle behavesas the disorder properties are altered and in particularthe contact angle hysteresis due to the chemical hetero-geneities.

We first focused on the possible correlations betweenapparent contact angles at both walls as a consequenceof the nonlocal effects of the interface dynamics. By im-posing a homogeneous top wall and a disordered bottomwall, we investigated how the dynamics of the top contactline is influenced by the bottom contact line fluctuations.Our results show that when the wetting properties at oneof the walls are different from those applied at the otherwall, both contact angles are coupled to each other: fluc-tuations above the mean at one wall produce fluctuationsbelow the mean at the other wall, and vice versa. On theother hand, when both walls have the same mean wettingproperties, this coupling effect is not observed. This anal-ysis then suggests that nonlocal effects may be relevanton the dynamics of the contact line, so that any analyt-ical and/or numerical approaches of contact line motionin microchannels and more general confined geometries,should take them into account.

On the other hand, a careful statistical study of theadvancing and receding motion of the contact line has re-vealed that the apparent contact angle suffers a hysteresisbehavior induced by the disorder of the walls. More pre-cisely, we have shown that the value of the apparent con-tact angle in the limit of zero velocity depends on whetherthe contact line is advancing or receding, and it is char-acterized by a jump discontinuity, ∆θ, that depends onthe disorder strength of the chemical heterogeneities ofthe system. In particular, we have shown that such hys-teresis behavior is enhanced with the disorder strength.It is important to emphasize that our results show howa purely diffusive process can generate a variation of thecontact angle with the meniscus velocity including com-plex phenomena like hysteresis behaviour and/or nonlo-cal dynamics. This is also an indication that the abovephenomena are generic and not linked to any hydrody-namics.

Of particular interest would be the extension of thepresent study to topographically heterogeneous sub-strates, which also have a significant effect on the wet-ting characteristics of the solid-liquid pair (e.g. Refs. 37–39). Furthermore, as also noted in the Introduction, the2D system we considered can be viewed as a simplifiedversion of a 3D system and serving as a first step to-wards characterizing contact line dynamics in microchan-nel chemically (or topographically) disordered in boththe streamwise and spanwise direction. In this case, inaddition to nonlocality and contact angle hysteresis weexpect a number of phenomena and complexities whichare likely to play a crucial role in the advancement of theliquid into the microchannel, such as kinetic roughening,pinning–depinning motion or avalanche dynamics of thecontact line14,16. We hope to examine these and relatedproblems in future studies.


10

Appendix A: Model B in the limit of small Peclet number

In this Appendix we point out under which conditionsthe phase field model we use in the present study, which isdescribed in Sec. II B, can be obtained from other existingmodels that describe contact line dynamics.

We focus here on a regime where diffusion processesdominate over advection processes. It is important toemphasize, however, that in general both mechanismscan be important, and in order to take them into account,Eq. (13) describing the dynamics of the phase field has tobe replaced by the following advectiondiffusion equation:

∂φ

∂t+ v ·∇φ = ∇ ·M∇µ, (A1)

where v represents the bulk velocity of the fluid. Theabove equation is then coupled to the Navier-Stokes equa-tion for the velocity field:

ρdv

dt= −∇p + η∇2v + µ∇φ, (A2)

where ρ and η are the fluid density and viscosity, re-spectively, and p represents the pressure. The model de-scribed by the above Eqs. (A1) and (A2) is usually re-ferred to as model H (if we follow the Hohenberg andHalperin terminology19) and, when solid-fluid interac-tions are also taken into account via a surface energyterm (see Eq. 5), it has been used to model several sortof wetting situations, such as droplet motion7,31,35, orthin film dynamics40,41 (see also Refs. 25, 27, 28, and 30for general aspects on this model).

By using now the dimensionless variables defined inEq. (7) we can rewrite equation (A1) as:

∂φ

∂t= C2

h(∇ ·m∇µ− Pe v ·∇φ), (A3)

where we have used the diffusion time scale ζ2/M0α tonon-dimensionalize time, and the Peclet number has beendefined as:

Pe =V Lx

M0α, (A4)

with the mean interface velocity V being used as thecharacteristic velocity of the system. We can thereforesee that in the limit of Pe → 0, the above equation (A3)becomes completely decoupled from the Navier-Stokesequations (A2), obtaining a pure diffusive regime wherethe equation describing the contact line dynamics is givenby Eq. (15) only.

ACKNOWLEDGMENTS

We are grateful to the anonymous referee for insight-ful comments and suggestions. We thank Julia M. Yeo-mans for fruitful discussions. We acknowledge the finan-cial support of F.R.S.-FNRS (Belgian National Fund for

Scientific Research), EU-FP7 ITN Multiflow and ERCAdvanced Grant No. 247031.

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