Interfaces in inhomogeneous media:pinning, hysteresis, and facets

Will Feldman

September 24th, 2019

Liquid drops on rough surfaces

J. Bico et al.: Rough wetting 215

Fig. 1 Fig. 2

Fig. 1 – Rough surface in contact with a reservoir. A small displacement dz of the imbibition frontof height z is considered.

Fig. 2 – Microstructured surface with regular micronic spikes used for the experiment. Surfaceparameters can accurately be estimated from the picture. We get r = 1.3 and φS = 5%.

where γSL, γSV and γ are, respectively, the solid/liquid, solid/vapor and liquid/vapor inter-facial tensions. The first term in eq. (1) is related to the replacement of a dry solid by a wetone. It is proportional to the wetted area, i.e. to the factor (r − φS). The second term isless usual in impregnation processes: it is related to the creation of a liquid/vapor interfaceassociated with the film propagation. Note that gravity was ignored in (1), which correspondsto textures of small size (much smaller than the capillary length) and small heights. Thegravity-limited case is treated in the appendix.

The liquid should rise if dE is negative. Introducing Young’s law (γ cos θ = γSV − γSL)gives, as a condition for imbibition,

θ < θc with cos θc =1 − φS

r − φS, (2)

where θ is the equilibrium contact angle of the liquid on an ideal flat surface of the samechemical composition.

Criterion (2) (logically) appears as intermediate between wetting and wicking criteria. Fora flat surface (r → 1), the surface is wetted if the contact angle reaches 0 (θc = 0), whilea porous medium (r → ∞) is invaded for liquids having a contact angle smaller than π/2(θc = π/2). More generally, since we have r > 1 and φS < 1, eq. (2) always defines a criticalcontact angle intermediate between 0 and π/2. Thus, the ability of a textured surface to drivea liquid can be tuned by its surface design. For a given surface composition and liquid (i.e.,fixing θ), the nature of the texture (which determines r and φS) decides if condition (2) issatisfied, or not. In a general case (disordered surfaces), the parameters r and φS are deeplyintricate (and φS may depend on θ). However, the use of micropatterned surfaces allows todecouple these two parameters, and even to treat them as independent, which is now discussedusing such a model surface.

216 EUROPHYSICS LETTERS

Fig. 3 Fig. 4

Fig. 3 – Pinning of the contact lines on the corners of the crenellations which ensures a constant valuefor the surface fraction φS.

Fig. 4 – Metastable square drop of ethyl malonate (θ = 32◦) on the structured surface. The contactline follows the directions of the rows.

Surfaces with a controlled design. – Figure 2 shows a surface achieved by Marzolin bycasting a sol-gel silicate on a silicon wafer [8]. It consists in spikes (typical size one micron)which are regularly spaced. We used this model surface to test a possible invasion of thetexture by various liquids.

For having a partial wetting with oils, octadecyl trichlorosilanes were grafted on the sample.A planar silicon wafer where the same molecules were grafted was used as a reference surface,allowing us to measure the advancing contact angle θ. The measurements were done opticallywith an accuracy of 5◦ and the contact angle hysteresis on these surfaces was found to besmall (< 10◦).

Then, experiments similar to the one sketched in fig. 1 were realized with different liquids(i.e. various contact angles). The sample was partially immersed in a reservoir of liquid, andthe existence of an invasion (or not) was monitored, together with the dynamics of the rise.The film was easily detected thanks to the darkening it induced on the sample. Our majorresult is the observation of an invading film climbing up to the top of the sample (typicallyone centimeter) for liquids having a small contact angle. A threshold in contact angle wasindeed observed, above which the invasion was not observed. The value of this threshold wasfound to be between 30◦ and 35◦.

The surface roughness r is easy to estimate from SEM (scanning electronic microscopy)images and is r = 1.30 (±0.05). The solid fraction φS on such a surface should be independentof the contact angle θ, because of the possibility for the contact lines to pin on the corners ofthe crenellations (fig. 3) [9]. If the angle is larger than 0, the top of the spikes remains dry;in the same time, an angle smaller than π/2 makes favorable the filling of the texture by theliquid. In such conditions, the parameter φS is just the ratio of the area of the top of thespikes over the total area of the sample, which is 5% here [10]. Then, relation (2) predicts avalue of θc equal to 40◦.

This value is close to the observed one—but we consider as significant the small differencebetween both. Criterion (2) is thermodynamic, and metastable states can also be achievedbecause there are activation barriers to jump for the contact line. This is related to thedisconnected nature of the texture: if we easily understand how a wetting liquid can invadea continuous groove in a solid [11], it is less obvious to figure out how the contact line pinnedon a row of spikes can “know” that there is another row ahead. (Thus, hysteresis can occur:

Marzolin, Smith, Prentiss and Whitesides Adv. Mater. (1998)Bico, Tordeaux and Quere Euro. Phys. Lett. (2001)

Scaling limit of boundary sandpile model

Simulations from Smart and F., ARMA ‘18Model introduced by Aleksanyan and Shahgholian.

A model free boundary problem

Both pictures can be explained by a free boundary problem of thefollowing form:{

∆u = 0 in {u > 0} and

|∇u| ∈ [Q∗(nx),Q∗(nx)] on ∂{u > 0}.Facets caused by discontinuities of endpoints of the pinninginterval [Q∗(nx),Q∗(nx)], as a function of the normal direction(idea goes back to Caffarelli and Lee ‘07).

I Derived as scaling limit of boundary sandpile model (Smartand F., ARMA ‘18).

I Derived as homogenization limit of continuous free boundaryproblem (F., preprint ‘18):

∆uε = 0 in {uε > 0} and |∇uε| = Q( xε ) on ∂{uε > 0}.

Inhomogeneity modeled by Q > 0 and Zd -periodic.

Structure of the effective PDE

Smart and F., ARMA ‘18

Structure of the effective PDE

Theorem (Smart and F., ARMA ‘18)

Define S : 2πTd → R by S(θ) = − log(1 + 1d

∑dj=1 cos θj), and let

S : Zd → C be the corresponding Fourier transform. Then S isreal and positive on Zd and for all e ∈ Sd−1,

Q∗(e) =1√2d

exp

(12

∑k∈Zd : k·e=0

S(k)

).

Where Q∗(e) is the upper endpoint of pinning interval associatedwith boundary sandpile scaling limit.

Theorem (F., preprint ‘18)

(Informally) Similar qualitative continuity properties for continuouscase in d = 2.

Minimal and maximal solutions

Minimal (and maximal) solutions play a key role:{∆u = 0 in {u > 0} and

|∇u| = Q∗(nx) on ∂{u > 0}.

New theory needs to be developed due to the discontinuous freeboundary condition.

Theorem (Smart and F., ARMA ‘18)

Strict comparison principle holds for

∆u = 0 in {u > 0} with |∇u| = Q∗(nx) on ∂{u > 0}

when d = 2 or in arbitrary dimension and convex setting.

Future Directions / Open Questions

I Mathematical follow-up questions:I General comparison principle and explaining facet shapes in

d ≥ 3?I Optimal regularity of the free boundary for the discontinuous

free boundary condition?I Presence of facets with co-dimension ≥ 2?

I Energy based approach, perhaps via dissipative evolutions,volume constrained solutions.

I Generic discontinuities of pinning interval in continuousmodel?

I What phenomena need to be explained with rough surface (asopposed to chemically patterned)?

I Random media. . .

Thank you for your attention!

Moving interfaces in random media

Forced mean curvature flow

Interface Γt evolving by normal velocity with planar initial datae · x = 0 (outward normal e)

Vn = −κ+ c(x) + F .

Here κ is mean curvature, c(x) is inhomogeneous environment,constant F is large scale external driving force (e.g. pressure,contact angle, or magnetic field).

Model for

I Flow in porous media

I Contact line motion

I Domain boundaries in magnetic materials

Pinning and depinning

Expectation: there is a pinning interval [F∗(e),F ∗(e)]

F ∗(e) = inf{F : lim inft→∞

u(0,t)t > 0}.

Front has positive speed outside of the pinning interval. Can alsodefine the depinning transition value

F d(e) = inf{F : limt→∞

u(0, t) = +∞ a.s}

Open questions:

I Are the depinning and positive speed transitions the same?

I Is the propagating interface flat for F > F ∗?

I What is the behavior of c(F ) near the depinning threshold?Conjectured universality c(F ) ∼ (F − F ∗)θ.

Propagation as a flat front

Take now F = 0. Periodic media:

I (Lions and Souganidis, ‘05) Lipschitz estimates and existenceof correctors under the coercivity condition

infRd

[c(x)2 − (d − 1)|Dc |] > 0.

I (Caffarelli and Monneau, ‘14) Counter-example in d ≥ 3,homogenization in d = 2 without a Lipschitz estimate withweaker coercivity

infRd

c > 0.

Random media:

I (Armstrong and Cardaliaguet, ‘15) Homogenization in d ≥ 2with the L-S coercivity condition (finite range).

I (F., in preparation ‘19) Homogenization in d = 2 with C-Mcondition, counter-example in d ≥ 3.

Liquid drops on rough surfaces

(yt¼ 1s4yt¼ 2s). The advancing contact line subsequentlybecomes pinned at the next row of pillars (t¼ 3 s) and theapparent contact angle continues to increase (t¼ 4 s) until itsatisfies the local depinning criteria similar to that at t¼ 1 s. As aresult, despite the intrinsic hydrophilic (o90!) behaviour of theflat surface, the corresponding rough surface demonstrated anapparent hydrophobic behaviour where the advancing contactangles exceeded 90!. This observation contradicts the classicalWenzel equation, which predicts that the degree of hydrophilicity(hydrophobicity) of a surface increases upon increasing theroughness19. However, the Wenzel equation does not capture theadvancing behaviour of these droplets since it does not accountfor the role of the local contact line pinning in the advancingstage (Fig. 2c). During a typical advancing stage, the portion ofthe contact line on the top surface of a micropillar ‘1’ advanceswith the local contact angle equal to the intrinsic contact angle onan equivalent flat surface (yF,A). Once the advancing frontapproaches the pillar edge, the contact line becomes pinned ‘2’and the local contact angle is required to exceed yF,Aþ 90!, ‘3’ forthe front to advance further along the pillar sidewalls ‘4’. The highenergy barrier associated with the local depinning contact anglesof yF,Aþ 90! results in an apparent global pinning (everywherealong the contact line) and the unusually high advancing contactangle values on these otherwise hydrophilic (yF,Ao90!) surfaces.

Estimation of advancing contact angle. The contact angleof a growing Cassie–Baxter droplet yCB,A on a chemically pat-terned surface with circular hydrophobic defects or on a super-hydrophobic surface (Fig. 3a) was previously32 shown to beinfluenced by local contact line distortion and modelled asfollows:

cosyCB;A ¼DL

cosy1;Aþ 1# DL

! "cosy2;A ð1Þ

where y1,A and cosy2,A are the corresponding intrinsic advancingcontact angles of the two background surfaces comprising the

heterogeneous surface. Now, accounting for the contact linepinning at the pillar edges (Fig. 2c) on the surfaces investigated inthis study where the local contact angle is required to exceedyF,Aþ 90! for the front to advance further along the pillarsidewalls (Fig. 3b), the apparent advancing contact angle of theWenzel-type droplets in our study can be computed by replacingy1,A by yF,Aþ 90! and y2,A by yF,A

cosyA;0! ¼DL

cosðyF;Aþ 90!Þþ 1# DL

! "cosyF;A ð2Þ

The subscript 0! indicates the direction of the line-of-sight for theside-view camera that, in this case, was parallel to the pillar rows(Fig. 2a). The experiment detailing the advancing, contact linepinning and receding modes of the droplet shown in Fig. 2 is alsoshown in Supplementary Movie 1.

The top- and side-profile visualization of the entire micro-scopic (diameterE200 mm) droplet during the contact linepinning stage highlights an interesting asymmetry in the contactangles along the two principle axes of symmetry on the surface.The droplets in Fig. 4 were formed on a surface comprised of asquare array of cylindrical micropillars with diameter D¼ 7 mm,pitch L¼ 20mm and height H¼ 13 mm. The images in Fig. 4a,bwere acquired during two different but highly repeatableexperiments, where the only difference was the line-of-sight forthe side-profile visualization of the droplet. The side-profileimages in Fig. 4a were captured such that the line-of-sight wasparallel to the pillar rows (0! line-of-sight, first axes of symmetry)while the side-profile images in Fig. 4b were capturedcorresponding to a line-of-sight of 45! (second axes of symmetry)with respect to the pillar rows. Top-view images at t1 right afterthe end of the advancing stage are shown on the top in Fig. 4a,b,as they could not be obtained during the advancing stage due tothe obstruction from the piezo inkjet head used for liquidaddition (see Supplementary Movie 2). Comparison of thecorresponding side-profile images show an interesting anisotropyin the contact angle, where the apparent contact angle along the

Global pinning+

evaporation

Octagon

Square

Rectangle Hexagon

Regular hexagon

Dodecagon

102°

109°

71°

90°

Figure 1 | False-colour images of various droplet shapes and patterns on micropillar surfaces. (a) Side- and top-view images of the droplet. The imageson the left correspond to an advancing stage where the apparent contact angle is 490! and the actual contact line is not visible. The actual octagonalcontact line is visible in the image on the right once the droplet evaporates in a global pinning mode and the apparent contact angle is o90!. The twoside-view images acquired along the line-of-sight of 45! (second axes of symmetry, green dotted line) with respect to the pillar rows have beenartificially rotated by 45!. The side-view images along with contact angles. (b) Different droplet shapes obtained by varying cylindrical micropillar arraygeometry. (c) The ability to pattern an array of microscopic droplets with excellent spatial control. Scale bar, 100mm and the geometry is defined inSupplementary Table 1.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5975 ARTICLE

NATURE COMMUNICATIONS | 5:4975 | DOI: 10.1038/ncomms5975 | www.nature.com/naturecommunications 3

& 2014 Macmillan Publishers Limited. All rights reserved.

Raj, Adera, Enright and Wang Nat. Commun. (2014)

Flow in lattice porous mediumSA82 R Lenormand

Figure 5. Imbibition. ( U ) Injection of oil (black) displacing air in a squarc network with a narrow pore size distribution. ( b ) Injection of oil (black) displacinb air in a triangdar net.work with a narrow pore size distribution. (c) Injection of oil (black) displacing air in a square network wit.h a wide pore size distribution.

3. Physical mechanisms at tlie pore level

Experiments show that both fluids can flow simultaneously in the same channel with different velocities, the wetting fluid remaining in the extreme corners of the cross section and roughness of the walls (figure 1). This effect explains the richness of mechanisms involved during displacements.

A displacement can be divided into three parts: (i) flow of the injected fluid from the entrance towards the moving meniscus, (ii) displacement of the meniscus, and (iii) flow of the displaced fluid towards the exit.

3.1. Menisciis displacements

The mechanisms are different in drainage and imbibition.

R Lenormand J. Phys.: Condens. Matter (1990)