Geometry-induced interface pinning at completely wet walls
GGeometry-induced interface pinning at completely wet walls
Alexandr Malijevsk´y
Department of Physical Chemistry, University of Chemical Technology Prague, Praha 6, 166 28, Czech Republic;Department of Molecular and Mesoscopic Modelling,ICPF of the Czech Academy Sciences, Prague, Czech Republic
We study complete wetting of solid walls that are patterned by parallel nanogrooves of depth D andwidth L with a periodicity of 2 L . The wall is formed of a material which interacts with the fluid via along-range potential and exhibits first-order wetting transition at temperature T w , should the wall isplanar. Using a non-local density functional theory we show that at a fixed temperature T > T w theprocess of complete wetting depends sensitively on two microscopic length-scales L + c and L − c . If thecorrugation parameter L is greater than L + c , the process is continuous similar to complete wettingon a planar wall. For L − c < L < L + c , the complete wetting exhibits first-order depinning transition corresponding to an abrupt unbinding of the liquid-gas interface from the wall. Finally, for L < L − c the interface remains pinned at the wall even at bulk liquid-gas coexistence. This implies that nano-modification of substrate surfaces can always change their wetting character from hydrophilic intohydrophobic, in direct contrast to the macroscopic Wenzel law. The resulting surface phase diagramreveals close analogy between the depinning and prewetting transitions including the nature of theircritical points. The recent advances in nanophysics have not only re-vealed promising possibilities in modern technologies butalso induced new theoretical challenges. This includes aparticularly important problem that has attracted enor-mous interest across different scientific branches andwhich can be formulated as follows: what is the effectof a solid surface structure on its wetting properties? Ona macroscopic level, the influence of a non-planar struc-ture on adsorption behaviour of the substrate can be de-scribed by Wenzel’s law [1]cos θ ∗ = r cos θ , (1)which relates Young’s contact angle θ of a liquid dropleton a planar surface with an apparent contact angle θ ∗ of a liquid droplet on a structured surface. Since theroughness parameter r >
1, Eq. (1) implies that θ ∗ > θ if θ > π/ θ ∗ < θ if θ < π/ T w (and at bulk liquid-gas coexis-tence) which corresponds to vanishing of the contact an-gle, θ ( T w ) = 0, the corrugated substrate also exhibitscontinuous wetting transition at the same temperature.The character of the wetting transition is also unchanged FIG. 1: Sketch of a cross-section of the model substrate withmicroscopic grooves of depth D and width L . The system isassumed to be periodic along the x -axis with a periodicity of P = 2 L and translation invariant along the y -axis. if the planar substrate exhibits first-order wetting butin this case the wetting temperature of the corrugatedsubstrate is shifted towards lower values, qualitatively inline with Eq. (1) [18, 20, 21]. Moreover, the wetting canbe preceded by unbending (or filling) transition corre-sponding to an abrupt condensation of the fluid insidethe wall troughs, provided the corrugation amplitude issufficiently large.These predictions are based on a mesoscopic analysisof the liquid-gas interface interacting with the solid wallaccording to an effective (binding) potential [25]. How-ever, if the substrate structure is microscopic, i.e. on thescale of molecular diameters, a more detailed treatmentof an adsorbed fluid is needed. In this work, we consider amodel substrate formed by a solid planar wall into whicha one-dimensional array of rectangular grooves of depth D and width L is etched with a periodicity P = 2 L . Thegroove parameters D and L are deemed to be microscop- a r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r FIG. 2: A comparison between surface phase diagrams for aplanar wall exhibiting first-order wetting transition at tem-perature T w (top left) and a periodically corrugated wall ata fixed temperature T > T w (bottom left). Also shown arethree adsorption isotherms corresponding to thermodynamicpaths denoted in the phase diagrams. ically small, while the length of the grooves L y alongthe remaining Cartesian axis is assumed to be macro-scopically long, so that the wall corrugation breaks thetranslation symmetry of the system only in one direction(cf. Fig. 1). The wall is at contact with a bulk gas at a(subcritical) temperature T > T w and the chemical po-tential µ < µ sat ( T ), where T w is the wetting temperatureat which first-order wetting transition occurs at the cor-responding planar wall and µ sat ( T ) is the chemical poten-tial at bulk liquid-gas coexistence. For planar walls theprocess µ → µ sat ( T ) is known as complete wetting andcan be characterized by a an unbinding of the liquid-gasinterface mean height (cid:96) which eventually diverges accord-ing to the power-law (cid:96) ∼ | δµ | − β co where δµ = µ − µ sat and the critical exponent β co = 1 / T w extends to T > T w and below µ sat , giving rise to a finite jump in (cid:96) ( µ ); this prewet-ting transition terminates at the surface critical point T sc . Schematically, the surface phase diagram for com-plete wetting at a planar wall is displayed in Fig. 2, to-gether with three illustrative adsorption isotherms. Herewe also show the L - δµ phase diagram corresponding toour model substrate which summarizes the main resultsof this work. It demonstrates three possible adsorptionscenarios, depending on L : i) the adsorption is continu-ous similar to complete wetting on a planar wall if thecorrugation parameter L is greater than a certain criti-cal value L + c ; ii) for L below L + c but above L − c the ad-sorption exhibits first-order depinning transition at somevalue µ ( L ) < µ sat corresponding to an abrupt depinningof the interface from the wall followed by a continuousdivergence of the interface height as µ → µ sat ; iii) finally, for L < L − c the interface remains pinned at the wall evenat µ sat preventing complete wetting despite Young’s con-tact angle θ = 0.We have obtained our results using a microscopic den-sity functional theory (DFT) [28], which has proven to bean extraordinary useful tool for a description of structureand phase behaviour of inhomogeneous fluids [29]. Allthe information about the given model fluid propertiesis embraced in the intrinsic free energy functional F [ ρ ]of the local one-body fluid density ρ ( r ) which, for simplefluids, can be decomposed in a perturbative manner asfollows: F [ ρ ] = F id [ ρ ] + F rep [ ρ ] + F att [ ρ ] . (2)Here, F id = β − (cid:82) d r ρ ( r ) (cid:2) ln( ρ ( r )Λ ) − (cid:3) is the ki-netic (ideal gas) contribution to the free energy where β = 1 /k B T is the inverse temperature and Λ is thethermal de Broglie wavelength. The excess part ofthe free energy functional due to the fluid-fluid in-teractions is further separated to the repulsive por-tion F rep , which is mapped onto a system of hardspheres with a diameter σ within the non-local Rosen-feld fundamental-measure-theory functional [30], and theattractive part which we treat in the mean-field manner F att = (cid:82) (cid:82) d r d r (cid:48) ρ ( r ) ρ ( r (cid:48) ) u att ( | r − r (cid:48) | ). For the attrac-tive part of the fluid-fluid interaction, u att ( r ), we havechosen the truncated and non-shifted Lennard-Jones-likepotential u att ( r ) = r < σ , − ε (cid:0) σr (cid:1) ; σ < r < r c , r > r c , (3)where the parameters ε and σ are used as the energy andlength-scale units, respectively, and where the potentialcut-off is set to r c = 2 . σ . The microscopic model ac-counts accurately for the short-ranged fluid correlations(and thus the packing effects) and satisfies exact statisti-cal mechanical sum rules [31]. The confining wall (illus-trated in Fig. 1), steps into the theory within the externalpotential V ( r ) = V ( x, z ) which is obtained by integrat-ing the wall-fluid atom-atom interactions φ w ( r ) over thewhole volume of the wall. With the wall atoms assumedto be distributed uniformly with a density ρ w and inter-acting with the fluid atoms via the Lennard-Jones (LJ)potential φ w ( r ) = 4 ε w (cid:104) ( σ/r ) − ( σ/r ) (cid:105) , the wall po-tential can be expressed as V ( r ) = V π ( z ) + ∞ (cid:88) n = −∞ V D ( x + 2 nL, z ) , (4)except for the region corresponding to the domain of thewall in which case V ( x, z ) = ∞ as the wall is impenetra-ble. The potential V D ( x, z ) of a single pillar of height D and width L can be split into the repulsive and attractivecontributions V D ( x, z ) = V ( x, z ) + V ( x, z ) where V ( x, z ) = − π ε w σ ρ w [ ψ ( x, z ) − ψ ( x, z − D ) − ψ ( x − L, z ) + ψ ( x − L, z − D )] (5)and V ( x, z ) = πε w σ ρ w [ ψ ( x, z ) − ψ ( x, z − D ) − ψ ( x − L, z ) + ψ ( x − L, z − D )] (6)with ψ ( x, z ) = 2 x + x z + 2 z x z √ x + z (7) and ψ ( x, z ) = 1128 128 x + 448 x z + 560 x z + 280 x z + 35 x z + 280 x z + 560 x z + 448 z x + 128 z z x ( x + z ) / − z . (8) d m / e L / s FIG. 3: The surface phase diagram as obtained from DFT for T = 0 . T c and D = 10 σ . The line displays the loci of thefirst-order depinning transition and terminates at the criticaldistance L + c ≈ σ above which the transition is continuous.Below L − c ≈ σ the gas-liquid interface remains pinned at thewall (which is thus not wet) even at saturation δµ = 0 rep-resented by the horizontal line. Also shown are the spinodallines (dashed) which denote the limit of stability of the de-pinned state (the lower one) and the pinned state (the upperone). The symbols × (+) indicate the points for which thecorresponding binding potentials are plotted in Fig. 5 (Fig. 6). Finally, the potential V π ( z ) in (4) is the standard 9-3 LJpotential induced by a planar wall spanning the volume z < ρ ( r )] = F [ ρ ( r )] + (cid:90) ρ ( r )( V ( r ) − µ )d r (9)which is solved iteratively on a two-dimensional grid witha uniform spacing of 0 . σ . The minimization determinesthe equilibrium density profile ρ ( r ) and also the thermo- FIG. 4: Equilibrium density profiles of pinned (left) and de-pinned (right) coexisting wetting states for L = 5 σ (upperpanels) and L = 10 σ (bottom panels). dynamic grand potential Ω = min Ω[ ρ ( r )]. Having set thewall parameter ε w ρ w = ε/σ we find the wetting temper-ature of the planar wall T w ≈ . T c , where T c is the bulkcritical temperature of the fluid.In Fig. 3 we display the surface phase diagram in the L - δµ projection obtained from the DFT for T = 0 . T c >T w and the grooves depth D = 10 σ . The phase dia-gram shows the line corresponding to first-order depin-ning transition at which the grand potentials obtained byminimizing of (9) for the pinned (low adsorption) and de-pinned (high adsorption) states are equal. The line con-nects the (horizontal) bulk coexistence line at L − c ≈ σ meaning that below this threshold the wall structure pre-vents complete wetting. The depinning line terminatesat the critical point L + c ≈ σ above which the depin-ning is continuous. For L > L + c , a unique solution for L = 1 1 s W s / e L = 8 s W s / e l / s l / s L = 5 s W s / e l / s FIG. 5: Binding potentials for (a) L = 5 σ , (b) L = 8 σ and (c) L = 11 σ , corresponding to the coexistence of thepinned and the depinned state. In all the cases, the bindingpotential exhibits two local minima of the same depth whichessentially merge when L = 11 σ indicating a close proximityto the critical point L + c . the density profile is always obtained regardless of theinitial state which the minimization of (9) starts from.However, below L c , the function Ω( L ) exhibits two lo-cal minima within the interval constrained by the spin-odal lines which are also shown in Fig. 3. Therefore, thisinterval indicates a range of metastable extensions of ei-ther solution characteristic to first-order transitions. Thespinodals display the loci where one of the two minimain Ω( L ) vanishes and becomes an inflection (cf. Fig. 6below), i.e. the limit of stability of the corresponding(pinned or depinned) configuration.To illustrate the change in the structure of the fluidat the depinning transition we show in Fig. 4 densityprofiles (over three periods) corresponding to coexistingpinned and depinned states for L = 5 σ and L = 10 σ .The two examples differ rather remarkably; for low L ,the depinned state possesses a thick wetting layer withessentially flat interface, while the upper part of the wallis only microscopically wet in the pinned state. For large L , the width of the wetting film after the transition issubstantially smaller and exhibits distinct periodic cor-rugation of the liquid-gas interface which follows closelythe lateral inhomogeneity in the wall potential. Beforethe transition, liquid droplets that are pinned at the walledges are now present, as the width of the pillars is largeenough to accommodate them. One should also noticethe strongly inhomogeneous fluid structure in the groovesshowing pronounced layering that get connected at thedepinned states; this suggests that the transition can alsobe viewed as bridging of the condensed phase filling thegrooves.Some more details about the depinning transitioncan be obtained by constructing the binding potential W ( (cid:96) ), i.e. the constrained free energy per unit length W s / e W s / e W s / e l / s L = 6 s l / s L = 7 s W s / e l / s L = 9 s l / s L = 1 0 s FIG. 6: Binding potentials for various values of L : (a) L = 6 σ , (b) L = 7 σ , (c) L = 9 σ and (d) L = 10 σ , as ob-tained from DFT at the fixed chemical potential µ = − . ε corresponding to the depinning transition for L = 8 σ . Forthe lowest ( L = 6 σ ) and the highest ( L = 10 σ ) value of thecorrugation parameter displayed, one of the two local minimadisappears suggesting that these states are already beyondthe spinodal points of the transition (cf. Fig. 3). at the fixed mean height of the adsorbed film (cid:96) (fromthe grooves bottom), W ( (cid:96) ) = min ρ ( r ) Ω[ ρ ( r )] | (cid:96) /L y , sothat the mean-field equilibrium state is given by theglobal minimum of W ( (cid:96) ). For this, we minimize thegrand potential (9) as a subject of fixed adsorption [32]Γ = (cid:82) d x (cid:82) d z ( ρ ( x, z ) − ρ b ) where ρ b ( µ ) is the bulk fluiddensity. In Fig. 5 we display the binding potentials cor-responding to the three points laying on the depinningline as depicted in Fig. 3. For the lowest value of L , thebinding potential possesses two distinct local minima ofthe same depths that are separated by a well pronouncedfree-energy barrier. On increasing L , i.e. by approaching L + c , the free-energy barrier is lowered, as well as the gapbetween the two minima which eventually merge at thecritical point. Clearly, this is only the second minimumof W ( (cid:96) ) the position of which depends on L , with the firstminimum being always located near D corresponding tothe grooves top.In Fig. 6 we also show the binding potentials for thepoints away of the depinning transition at the fixed chem-ical potential. The effect of varying L is now differ-ent; namely it determines the depths of the two com-peting free-energy minima but their locations remain un-changed within the whole interval of L between the spin-odal points (cf. Fig. 3) beyond which only one local min-imum in W ( (cid:96) ) exists. This can be explained as follows.Within the sharp-kink approximation [25], the bindingpotential of the system in the depinned state can be writ- l / s d m / e FIG. 7: DFT results for the height of the wetting layers atthe structured wall (black solid line) with D = 10 σ and L = 5 σ , and the planar wall (red dashed line) with the po-tential strength as given by Eq. (12). Also shown is the resultobtained by a direct solution of Eq. (12) (blue dotted line). ten as W ( (cid:96) ) = 2( (cid:96) − D ) Lδµ ∆ ρ + AL ( (cid:96) − D ) + AL(cid:96) ( (cid:96) > D ) . (10)Here, the first term on the right hand side is the free-energy cost for the presence of the metastable liquid (ig-noring the constant contribution due to the filled grooves)and the remaining terms account for the local effectiveinteraction between the wall and the liquid-gas interfacewhich is considered to be flat. This interaction is approx-imated by a simple quadratic power-law in the inverseheight of the interface with the amplitude (Hamaker con-stant) A = π ∆ ρρ w ε w σ /
3, where ∆ ρ is the difference be-tween the bulk coexisting densities. Therefore, within theapproximation W ( (cid:96) ) scales linearly with L and hence theinterface height is L -independent, in line with the DFTresults. It should be emphasized that the model (10) isonly applicable to the depinned state (and is meaningfulonly above the lower spinodal of the phase diagram) andhas no relevance to the nature of the depinning transi-tion. Indeed, the full model would rely on a definitionof two distinct fluid configurations (pinned and depinnedstates) and would not thus be able to predict the presenceof the critical point L + c . However, Eq. (10) can be fur-ther used to find a correspondence between our substratemodel involving grooves and a simple planar wall coveredby a wetting film of width (cid:96) π , for which the dominatingcontribution to the binding potential is W π ( (cid:96) π ) = δµ ∆ ρ(cid:96) π + A (cid:48) (cid:96) π , (11)defined per unit area. The effective Hamaker constant A (cid:48) of the planar wall is determined by equating the heights of the wetting films at the corresponding sub-strates, (cid:96) ( δµ ) = (cid:96) π ( δµ ), obtained by minimization of (10)and (11), respectively. This leads to the condition (cid:18) − (cid:96) (cid:19) = y (cid:96) − y , (12)for ˜ (cid:96) = (cid:96)/D as a function of the scaling parameter y =2 A/ ( δµ ∆ ρD ), which eventually yields A (cid:48) = ˜ (cid:96) A/y . Itcan be checked easily, that A (cid:48) → ∞ for D → ∞ and A (cid:48) → A for D →
0, as expected [33]. Using DFT, we test thisresult by comparing the height of the depinned interfaceat the structured wall with D = 10 σ and L = 5 σ and thepotential strength ε w = ε , with that corresponding to theplanar wall with the potential strength ε (cid:48) w /ε w = A (cid:48) /A as obtained from Eq. (12). The comparison shown inFig. 7 reveals that the two solutions, (cid:96) and (cid:96) π , are fairlyclose to each other, although the interface height abovethe planar wall (cid:96) π is systematically slightly larger. Forcompleteness, we also plot (cid:96) ( δµ ) as obtained directly fromEq. (12) which almost follows (cid:96) π ; the upshot is that whilethe asymptotic form of the planar binding potential (11)works very accurately within the displayed interval, thebinding potential for the structured wall (10) providesstill a reasonable approximation.In summary, we have studied adsorption of a solidwall structured by a linear array of parallel rectangulargrooves above the wetting temperature, so that Young’scontact angle of the wall is zero. We have found thatthe presence of the wall structure does not qualitativelychange the process of complete wetting unless the char-acteristic length of the wall structure L is microscopicallysmall. In this case, the mechanism of complete wettingis via bridging of liquid layers inside the grooves over thetop of the wall. The corresponding free energy changeinvolves a contribution associated with the line tension τ pertinent to a contact of the liquid-gas interface with thewall edges. This term competes within the excess free en-ergy with the surface tension effects as τ /L and is thus in-creasingly relevant as L decreases. Its role becomes domi-nant for L < L − c such that the free energy barrier cannotbe overcome even at the bulk coexistence δµ = 0 andthe bridging and complete wetting of the wall is thus notpossible. However, as our DFT calculations show, this isonly in the case when L is less than about four molecu-lar diameters ( σ ). For larger L , the system experiencescompetition between two free energy minima giving riseto first-order depinning transition. The edge effects andhence the transition is gradually weaker as L increasesand eventually becomes continuous at L = L − c ( ≈ σ )where the barrier vanishes. For L ≥ L + c the shape of theinterface changes smoothly all the way along its unbind-ing from the wall. The δµ - L phase diagram reveals someanalogy between prewetting and depinning transitions,although the curvatures of the corresponding lines haveopposite signs. This study can be extended in numer-ous ways. A natural generalization of our model wouldtreat the grooves width L and the periodicity P as in-dependent parameters; this, for example, would convertEq. (12) simply into (1 − / ˜ l ) = xy/ (˜ (cid:96) + ( x − y ) with x = L/P but the phase behaviour at such a wall canbe expected to be considerably more complicated. Fur-ther, here we have deliberately chosen a sufficiently hightemperature in order to avoid prewetting at the wall; for
T < T sc one expects competition between depinning andprewetting. It would also be interesting to explore D -dependence of the depinning phenomena and check pos-sible scaling properties as in Eq. (12). Finally, it shouldbe noted that in view of its pseudo-2D character andthe presence of long-range forces, the depinning transi- tion would not be rounded beyond the current mean-fieldanalysis due to thermal fluctuations and should thus beaccessible in real experiments. Acknowledgments
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