Wedge wetting by electrolyte solutions
aa r X i v : . [ c ond - m a t . s o f t ] S e p Wedge wetting by electrolyte solutions
Maximilian Mußotter ∗ and Markus Bier † Max Planck Institute for Intelligent Systems, Heisenbergstr. 3, 70569 Stuttgart, Germany andInstitute for Theoretical Physics IV, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany (Dated: October 15, 2018)The wetting of a charged wedge-like wall by an electrolyte solution is investigated by means ofclassical density functional theory. As in other studies on wedge wetting, this geometry is consideredas the most simple deviation from a planar substrate, and it serves as a first step towards morecomplex confinements of fluids. By focusing on fluids containing ions and surface charges, featuresof real systems are covered which are not accessible within the vast majority of previous theoreticalstudies concentrating on simple fluids in contact with uncharged wedges. In particular, the fillingtransition of charged wedges is necessarily of first order, because wetting transitions of chargedsubstrates are of first order and the barrier in the effective interface potential persists below thewetting transition of a planar wall; hence, critical filling transitions are not expected to occur forionic systems. The dependence of the critical opening angle on the surface charge, as well as thedependence of the filling height, of the wedge adsorption, and of the line tension on the openingangle and on the surface charge are analyzed in detail.
I. INTRODUCTION
Over the past few decades numerous theoretical andexperimental investigations have been performed aimingat a microscopic understanding of the phenomena of flu-ids at interfaces, e.g., capillarity, wetting, and spreading,which are of technological importance for, e.g., coatingprocesses, surface patterning, or the functioning of mi-crofluidic devices [1–5]. Particularly simple model sys-tems to investigate these phenomena theoretically areplanar homogeneous substrates, which have been stud-ied intensively [6–8]. This way, methods have been de-veloped to relate the thickness of fluid films adsorbed atsubstrates and the contact angle to fluid-fluid and wall-fluid interactions, to infer surface phase diagrams, and tocharacterize the order of wetting transitions.However, the preparation of truly flat homogeneoussubstrates requires a huge technical effort and in naturethere is no such thing as a perfectly flat surface [9]. Onthe one hand, one is always confronted with geomet-rically or chemically structured substrates, irregularly-shaped boundaries, or geometrical disorder. On the otherhand, modern surface patterning techniques allow for thetargeted fabrication of structured substrates with pits,posts, grooves, edges, wedges etc. in order to generatefunctionality, e.g., superhydrophobic surfaces [10]. Thisleads to the necessity of studying substrates beyond thesimple flat geometry, but the wetting properties of suchnonplanar substrates are very different from smooth andplanar walls and their description is much more complex.Perhaps the most simple of the aforementioned elemen-tary topographic surface structures are wedges, which areformed by the intersection of two planar walls meetingat a particular opening angle. First predictions of thephenomenon of the filling of a wedge upon decreasing ∗ [email protected] † [email protected] the opening angle have been based on macroscopic con-siderations [11, 12]. Microscopic classical density func-tional theory and mesoscopic approaches based on effec-tive interface Hamiltonians revealed that systems withlong-ranged Van-der-Waals interactions, where criticalwetting transitions of planar walls occur, exhibit crit-ical wedge filling transitions with universal asymptoticscaling behavior of the relevant quantities [13–15]. It hasbeen argued that the order of a filling transition equalsthe order of the wetting transition of a planar wall [16].However, it turned out later that the relation betweenthe orders of wetting and filling transitions is more sub-tle: If the wetting transition is critical then the fillingtransition is critical, too. Otherwise, if the wetting tran-sition is of first order then the filling transition may befirst-order or critical, depending on whether or not a bar-rier exists in the effective interface potential at the fillingtransition [17, 18]. A consequence of the latter scenariowith first-order wetting transitions is the possibility tohave first-order filling transitions, if the critical open-ing angle is wide, and critical filling transitions, if it isnarrow. These predictions from mesoscopic approacheshave been recently verified by microscopic classical den-sity functional theory [19, 20].In order to reduce complexity, all cited previous the-oretical studies on wedge wetting have been performedfor models of simple fluids. However, many fluids used inapplications, including pure water due to its autodissoci-ation reaction, are complex fluids containing ions, so thatthe generic situation of wedge wetting by electrolyte solu-tions is of enormous interest from both the fundamentalas well as the applied point of view. Despite the hugerelevance of electrolytes as fluids involved in wedge wet-ting scenarios [21], this setup has not been theoreticallystudied before on the microscopic level, probably due tothe expected lack of universality and increased complex-ity as compared to cases with critical wetting and fillingtransitions. Indeed, it turned out for planar walls thatthe presence of ions, not too close to bulk critical points,generates first-order wetting and a non-vanishing barrierin the effective interface potential below the wetting tran-sition [22]. Hence, on very general grounds, one expectsfirst-order filling transitions of wedges to take place forelectrolyte solutions.In the present work, a microscopic lattice modelis studied within a classical density functional theoryframework in order to investigate the properties of wedgewetting by electrolyte solutions. The usage of a latticemodel allows for technical advantages over continuummodels [22–24]. The model and the density functionalformulation is specified in Sec. II. In Sec. III first thebulk phase diagram and the wetting behavior of a pla-nar wall of the considered model are reported. Next,wedge wetting is studied in terms of three observables:the wedge adsorption, the filling height, and the line ten-sion. The dependence of these quantities on the wedgeopening angle, on the surface charge density of the wallsof the wedge, as well as on the strength and the range ofthe nonelectrostatic wall-fluid interaction are discussed indetail. Concluding remarks on the first-order filling tran-sition considered in the present work and the more widelystudied critical filling transition are given in Sec. IV. II. THEORETICAL FOUNDATIONSA. Setup
In the present work, the filling behavior of an elec-trolyte solution close to a wedge-like substrate is studied.Consider in three-dimensional Euclidean space a wedgecomposed of two semi-infinite planar walls meeting atan opening angle θ along the z -axis of a Cartesian co-ordinate system (see Fig. 1). Due to the translationalsymmetry in z -direction the system can be treated asquasi-two-dimensional. In between the two walls an elec-trolyte solution composed of an uncharged solvent (index“0”), univalent cations (index “+”), and univalent anions(index “-”) is present. The wedge is in contact with a gasbulk at thermodynamic coexistence between liquid andgas phase. This choice of the thermodynamic parametersallows for two different filling states of the wedge. Frommacroscopic considerations [11, 12], a critical opening an-gle θ C = π − ϑ (1)with the contact angle ϑ of the liquid can be derived,which marks the transition between the wedge beingfilled by gas (“empty wedge”) for θ > θ C and the wedgebeing filled by liquid for θ < θ C . It is of utmost impor-tance for the following to realize that, from the micro-scopic point of view, a macroscopically empty wedge istypically partially filled by liquid. θ ~r ~r v ~r v ′ ~r u ′ ~r u ~e u ~e u ′ ~s~r − ~s dd ~e z ~e y ~e x Figure 1:
FIG. 1. Schematic depiction of the studied system. The twounit vectors ~e u and ~e u ′ are parallel to the two walls whichmeet at the opening angle θ . An arbitrary location ~r can bespecified by the lateral and the normal components ~r u , ~r v or ~r u ′ , ~r v ′ with respect to the walls. The parallelogram close tothe wedge apex indicates the geometry of the unit cells bywhich the space in between the walls is tiled. Characterizing the dependence of the critical openingangle θ C on the wall charge and describing the partialfilling upon approaching the filling transition for θ > ∼ θ C are the objectives of the present study. B. Density functional theory
In order to determine the equilibrium structure of thefluid in terms of the density profiles of the three species,classical density functional theory [25] is used. As wet-ting phenomena typically require descriptions on sev-eral length scales, computational advantage is gained bystudying a lattice fluid model in the spirit of Refs. [22–24].In order to account for the special geometry of the systemat hand, the standard lattice fluid model is adapted byusing parallelograms as basic elements of the grid, whichis indicated by the parallelogram close to the apex of thewedge in Fig. 1. The size of an elementary parallelogram,which can be occupied by at most one particle of eitherspecies, is chosen such that, with d denoting the particlediameter, the sides parallel to the wall are of length d andthey are a distance d apart from each other (see Fig. 1).Each cell is identified by a pair ( l, j ) of integer indiceswhere l ≥ j represents the location parallel to the walls (see Fig. 1).The approximative density functional of this model usedin the present work can be written as β Ω[ φ ] = ρ max d X l,j h X α ∈{ , ±} φ α ; l,j (ln( φ α ; l,j ) − µ ∗ α + βV l,j ) + (1 − φ tot; l,j ) ln(1 − φ tot; l,j )+ 12 X n,m βU ∗ l,j ; n,m φ tot; l,j φ tot; n,m i + βU el , (2)where φ α ; l,j = ρ α ; l,j d denotes the packing fraction offluid component α ∈ { , ±} inside the cell specified bythe indices ( l, j ), φ tot = φ + φ + + φ − being the sum ofthe partial packing fractions, µ ∗ α is the effective chemicalpotential of component α , and ρ max = 1 / d is the max-imal number density of the fluid. In the following thevalues k B T = 1 /β with T = 300 K and ρ max = 55 . V l,j in Eq. (2) describes the non-electrostatic interaction of the wall with a particle in cell( l, j ). It is chosen to be independent of the specific par-ticle type. Here the wall-fluid interaction strength at agiven position ~r results from a superposition of interac-tions with all points ~s at the surface of the walls (seeFig. 1): βV ( ~r ) = Z ∞ d uβ Φ( | ~r − u~e u | )+ Z ∞ d u ′ β Φ( | ~r − u ′ ~e u ′ | ) , (3)where β Φ is the underlying molecular pair potential ofthe wall-fluid interaction. For the sake of simplicity theGaussian form β Φ( r ) ∼ exp (cid:18) − (cid:16) rλ (cid:17) (cid:19) (4)with decay length λ is used, which leads to the non-electrostatic wall-fluid interaction, Eq. (3), βV ( ~r ) = h exp (cid:18) − (cid:16) r v λ (cid:17) (cid:19) erfc (cid:16) − r u λ (cid:17) +exp (cid:18) − (cid:16) r v ′ λ (cid:17) (cid:19) erfc (cid:16) − r u ′ λ (cid:17) (cid:19) , (5)where the dimensionless coefficient h describes the wall-fluid interaction strength.The two remaining expressions in Eq. (2) considerthe interactions among the particles, which we consideras being composed of an electrically neutral molecularbody and, in the case of the ions, an additional chargemonopole. The way these interactions are treated regards the interactions as split in two contributions: the interac-tion between uncharged molecular bodies, which we referto as non-electrostatic contribution, and the interactionbetween charge monopoles. In the present work we ig-nore the cross-interactions between a charge monopoleand a neutral body. However the chosen model provesto be sufficiently precise as it qualitatively captures therelevant feature of an increase of the ion density for anincreasing solvent density. For example in the case of aliquid phase with density φ = 0 . φ = 0 . φ ± = 1 . · − in the gas to φ ± = 7 . · − in the liquid.In the Eq. (2), the non-electrostatic contribution tothe fluid-fluid interaction is treated within random-phaseapproximation (RPA) based on the interaction pair po-tential U ∗ l,j ; n,m between a fluid particle in cell ( l, j ) andanother one in cell ( n, m ). Here this interaction is as-sumed to be independent of the particle type and it isassumed to act only between nearest neighbors, i.e., be-tween particles located in adjacent cells.Finally, in Eq. (2) all electrostatic interactions, bothwall-fluid and fluid-fluid, are accounted for by the elec-tric field energy βU el . The electric field entering βU el is determined by Neumann boundary conditions set bya uniform surface charge density σ at the walls of thewedge, planar symmetry far away from the wedge sym-metry plane and global charge neutrality. Furthermore,the dielectric constant is assumed to be dependent on thesolvent density. It is chosen to interpolate linearly be-tween the values for vacuum ( ǫ = 1) and water ( ǫ = 80).This linear interpolation has been previously shown tomatch the behavior of the dielectric constant in mixturesof fluids very well [26]. In addition it is important tonote, that here the surface charge is not caused by thedissociation of ionizable surface groups, i.e., charge regu-lation as in Ref. [27] is not relevant here, but it is assumedto be created by an external electrical potential, whichis applied to the wall. One can imagine the wall beingan electrode with the counter electrode being placed farfrom the wall inside the fluid. C. Composition of the grand potential
Upon minimizing the density functional β Ω[ φ ] inEq. (2) one obtains the equilibrium packing fraction pro-files φ eq , which lead to the equilibrium grand potential -5-4-3-2-1 0.2 0.3 0.4 0.5 0.6 µ ∗ T ∗ C gasliquid FIG. 2. Bulk phase diagram in terms of the solvent chemicalpotential µ ∗ and the temperature T ∗ for fixed ionic strength I . The solid red line represents the liquid-gas coexistence linefor the salt-free case ( I = 0), which is given by the analyticalexpression µ ∗ = − T ∗ . The black crosses indicate points ofthe liquid-gas coexistence curve for the case I = 5 m M . Theshift is small, which also holds for all ionic strengths used inthis work (up to I = 100 m M ). β Ω eq = β Ω[ φ eq ] of the system. This equilibrium grandpotential can be decomposed into three contributions: β Ω eq = − βpV + βγA + βτ L. (6)The first contribution − pV with the pressure p andthe fluid volume V equals the bulk energy contribu-tion. It corresponds to the grand potential of an equally-sized system completely filled with the uniform gas bulkstate. The second term γA with the interfacial tension γ and the total wall area A corresponds to the quasi-one-dimensional case of the gas being in contact with a planarwall. The third contribution τ L with the line tension τ and the length L of the wedge is the only contributionto the total grand potential, where the influence of thewedge enters, and it is therefore of particular importancein the present work. III. RESULTSA. Bulk phase diagram
In the bulk region, far from any confinements, thedensities φ α , α ∈ { , ±} of the three fluid componentsbecome constant, and, due to local charge neutrality, φ + = φ − . This simplifies the density functional β Ω[ φ ]in Eq. (2), and the Euler-Lagrange equations read µ ∗ α = ln φ α − φ tot − T ∗ φ tot , (7) -0.03-0.02-0.0100.01 0 5 10 15 20 β ω d l (units of d) (a) φ t o t v (units of d) φ (gas)tot φ α φ + φ − (b) FIG. 3. (a) Effective interface potential βω as a function ofthe film thickness l of a liquid film between a planar wall andthe gas bulk phase and (b) equilibrium total packing fractionprofile φ tot ( v ), with v denoting the distance from the wall(see Fig. 1), corresponding to the minimum of βω ( l ) for ionicstrength I = 100 m M , temperature T ∗ = 0 .
43, wall-fluid in-teraction strength h = 0 . λ = 2 d, andsurface charge density σ = 0 . e/ d . The inset in panel (b)shows the corresponding ion packing fractions φ + and φ − asfunctions of the distance v from the wall. Panel (a) identifiesthe system exhibiting partial wetting for the present configu-ration. where 1 /T ∗ is proportional to the strength of the fluid-fluid interaction βU ∗ . For the ion-free case I = 0 theliquid-gas coexistence line is given by the analytical ex-pression µ ∗ = − T ∗ (see solid red line in Fig. 2). Forfixed but non-vanishing ionic strengths I the liquid-gascoexistence lines have been calculated numerically (seethe black crosses in Fig. 2). Whereas the deviations fromthe ion-free case are only marginal in the bulk phase dia-gram for all ionic strengths considered here, it is of majorimportance to determine the coexistence conditions pre-cisely, because surface and line properties (see Eq. (6))are highly sensitive to them. B. Electrolyte wetting on a planar wall
Before studying the filling behavior of a wedge, it isimportant to study the wetting of a planar wall becausethe results enter as the surface contributions to the totalgrand potential Eq. (6) and the quasi-one-dimensionalpacking fraction profiles provide the boundary conditionsfar away from the wedge symmetry plane. In the case ofa planar wall the density functional β Ω[ φ ] simplifies to aquasi-one-dimensional one and, due to the correspondingrelations r u = − r u ′ , r v = r v ′ (see Fig. 1), the expressionEq. (5) for the fluid-wall interaction becomes βV ( ~r ) = 2 h exp (cid:18) − (cid:16) r v λ (cid:17) (cid:19) . (8)With this set of equations one can determine the equilib-rium packing fraction φ α ; i of the fluid close to the planarwall, where the integer index i ≥ φ tot ] := ∞ X i =0 ( φ tot; i − φ (gas)tot ) (9)with the total packing fraction φ (gas)tot of the gas phaseat liquid-gas coexistence for the given temperature T ∗ ,which measures the additional amount of particles in ex-cess to the gas bulk phase due to the presence of the wall.Alternatively, one can consider the film thickness l [ φ tot ] := Γ[ φ tot ] φ (liquid)tot − φ (gas)tot (10)with the total packing fraction φ (liquid)tot of the liquid phaseat liquid-gas coexistence for the given temperature T ∗ ,which corresponds to the thickness of a uniform liquidfilm of packing fraction φ (liquid)tot with the same excess ad-sorption Γ[ φ tot ] as the equilibrium total packing fractionprofile φ tot .Minimizing the grand potential functional Eq. (2) fora planar wall (see Eq. (8)) with the constraint of fixedexcess adsorption Γ[ φ tot ], Eq. (9), or fixed film thickness l [ φ tot ], Eq. (10), and subtracting the bulk contributionof the grand potential as well as the wall-liquid and theliquid-gas interfacial tensions ( γ sl and γ lg , respectively),one obtains the effective interface potential βω [6]. Anexample for βω ( l ) is displayed in Fig. 3(a). The position l = l eq of the minimum of the effective interface potential βω ( l ) corresponds to the equilibrium film thickness. Thecorresponding equilibrium total packing fraction profile φ tot for the parameters chosen in Fig. 3(a) is shown inFig. 3(b).Using this procedure, one can determine the equilib-rium density profiles for different ionic strengths I , tem-peratures T ∗ , wall-fluid interaction strengths h , decaylengths λ , and surface charge densities σ . Figure 4(a) h C Γ h σ = 0 , λ = 1d
240 0 5 10 15 20 β ω d l (units of d) (a) Γ σ (units of e/ d ) σ C,λ =4 σ C, σ C, h = 0 . , λ = 4d h = 0 . , λ = 2d h = 0 . , λ = 1d (b) FIG. 4. Excess adsorption Γ (see Eq. (9)) for (a) an electri-cally neutral planar wall ( σ = 0) and decay length λ = 1 das function of the wall-fluid interaction strength h and (b)three different sets of wall-fluid interaction strength h anddecay length λ as function of the wall charge density σ . Bothpanels exhibit an increase of the excess adsorption Γ uponapproaching critical values h C or σ C , respectively, at whichthe system undergoes a wetting transition. The discontinu-ity of Γ at the critical values identifies the wetting transitionto be of first order. This can also be verified by consideringthe effective interface potential βω ( l ), as shown in the insetin panel (a) for conditions slightly above the wetting transi-tion. The barrier separating the local minimum at small filmthickness l from the global minimum at large film thickness lproves the first order nature of the wetting transition. displays the equilibrium excess adsorption Γ as functionof the wall-fluid interaction strength h for surface chargedensity σ = 0 and decay length λ = 1 d. Due to thevanishing surface charge, the packing fraction profiles ofthe two ion species are identical, φ + = φ − , hence thefluid is locally charge neutral and the electrostatic en-ergy βU el in Eq. (2) vanishes. Therefore, due to thesmall number densities of the ions, this case is similar toan ion-free system, where a wetting transition is caused σ c , σ c , θ C ( d e g ) σ (units of e/ d )empty λ = 2d λ = 1d filled FIG. 5. Critical opening angle θ C of the wedge, at which thefilling transition occurs, as function of the wall charge density σ for decay lengths λ ∈ { , } . For θ > θ C the wedgeis macroscopically empty, whereas for θ < θ C it is filled byliquid. The values θ C , derived via Eqs. (1) and (12), increasewith increasing wall charge σ . At wall charge density σ = σ C the critical angle of the filling transition is θ C = 180 ◦ , i.e., thefilling transition is actually the wetting transition of a planarwall (see Sec. III B). by an increase of the non-electrostatic wall-fluid interac-tion strength h (see Eq. (8)) up to a critical value h C .In contrast, Fig. 4(b) shows the excess adsorption Γ fordifferent sets of the wall-fluid interaction strength h andthe decay length λ as function of the surface charge den-sity σ . The values of h are chosen in such a way, thatthe three respective decay lengths λ =1 d, 2 d and 4 dlead to the same values of the volume integrals of thecorresponding wall-fluid interaction potentials, Z V d r βV ( ~r ) . (11)Here, the wall charge σ is varied and a wetting transitionis observed at a critical value σ C .All four setups in Fig. 4 exhibit the characteristics offirst-order wetting transitions, which are identified by fi-nite limits of Γ upon h ր h C or σ ր σ C . In additionfor all these cases the first-order nature has been verifiedby studying the effective interface potential (see inset inFig. 4(a)), which is clearly manifested by the energy bar-rier separating the local and the global minimum. For thequasi-ion-free case σ = 0 in Fig. 4(a) the choice Eq. (4) ofthe molecular pair potential of the wall-fluid interactionleads to a wetting transition of first order, in contrast tothe choice of the nearest neighbor potential in Ref. [23],which generates a second-order wetting transition. How-ever, it has been shown that for σ = 0 (see Fig. 4(b)) wet-ting transitions are of first order once the Debye lengthis larger than the bulk correlation length [22]. C. Wedge wetting by an electrolyte solution
Having studied the system under consideration in thebulk (Sec. III A) and close to a planar wall (Sec. III B),one can investigate wedge-shape geometries. As ex-plained in the context of Eq. (1), the system undergoesa filling transition for the opening angle θ (see Fig. 1)approaching the critical opening angle θ C from above.For θ < θ C the wedge is macroscopically filled by liq-uid, whereas for θ > θ C the wedge is macroscopicallyempty. In the following, the filling of an empty wedge,i.e., θ ց θ C , will be studied.Following Eq. (1), the critical opening angle θ C can becalculated from the contact angle ϑ of the liquid, whichis related to the depth of the minimum of the effectiveinterface potential by [6]cos ϑ = 1 + ω ( l eq ) γ lg (12)with the liquid-gas surface tension γ lg . Hence, the crit-ical opening angle θ C can be inferred from the wettingproperties of a planar wall using the method of Sec. III B.Figure 5 displays the critical opening angle θ C as func-tion of the wall charge σ for the case of decay lengths λ ∈ { , } . As the contact angle ϑ decreases upon in-creasing the wall charge due to the electrowetting effect[28], the critical opening angle θ C increases with increas-ing wall charge. For the critical wall charge σ = σ C thecritical opening angle θ C reaches the value of 180 ◦ , sincefor this wall charge the wetting transition of the planarwall occurs (compare Fig. 4(b)), i.e., for a planar wall thewetting and the filling transition are identical.Figure 6 displays the equilibrium packing fractionprofiles inside wedges with opening angles θ = 180 ◦ (Fig. 6(a)) and θ = 80 ◦ (Fig. 6(b)) with the parameters h , λ , and σ identical to those of Fig. 3(b). Away from thewedge symmetry plane the structure rapidly convergestowards that of a planar wall, which verifies the chosensize of the numerical grid being sufficiently large to cap-ture all interesting effects. Furthermore, the decrease ofthe opening angle, as shown in Fig. 6(b), leads to an in-crease of the density close to the tip of the wedge. Forexample the maximal density increases from 15 % of therelative density difference between liquid and gas densityto almost 30 %. However, the increase in the density islimited to the close vicinity of the tip of the wedge, whichis an indication of first-order filling transitions. In fact,in the presence of ions, wetting transitions at a planarwall are of first order with a barrier in the effective inter-face potential βω ( l ) (see Fig. 3(a)) being present for allstates below the wetting transition of a planar wall [22].Hence filling transitions of wedges are expected to be offirst order, too [17, 18].In order to describe the filling transition of a wedgequantitatively, several quantities have been studied. -15 -10 -5 0 5 10 15x (units of d)0510152025303540 y ( un i t s o f d ) (a) y ( un i t s o f d ) (b)-10 -5 0 5 10x (units of d) 00.050.100.150.200.250.30 ( φ t o t ( x , y ) − φ ( ga s )t o t ) / ( φ ( li q u i d )t o t − φ ( ga s )t o t ) FIG. 6. Equilibrium packing fraction profiles φ tot ( x, y ) inside a wedge with opening angle (a) θ = 180 ◦ and (b) θ = 80 ◦ for ionicstrength I = 100 m M , wall-fluid interaction strength h = 0 .
093 27, decay length λ = 2 d, and wall charge density σ = 0 . e/ d .Far away from the symmetry plane of the wedge the packing fraction profiles coincide with those at planar walls (see Fig. 3(b)).Upon decreasing the opening angle θ , an increase of the density close to the tip of the wedge occurs (see panel (b)). Firstly the wedge adsorption∆ = X i X j ( φ i,j − φ (gas)tot ) − Γ l wall /d, (13)with the length of the wall l wall shall be discussed. In thespirit of the excess adsorption Γ at a planar wall (Eq. (9)),this quantity ∆ measures the excess of an inclined wedgeabove the excess adsorption Γ of a planar wall. In Fig. 7the wedge adsorption ∆ is shown as function of the open-ing angle θ and of the wall charge density σ for decaylengths λ = 1 d (Fig. 7(a)) and λ = 2 d (Fig. 7(b)). Theionic strength is I = 100 m M and the wall-fluid interac-tion strength h has been chosen as in Fig. 4(b). Upondecreasing the opening angle θ the wedge adsorption ∆increases, regardless of the wall charge density σ , thedecay length λ , or the non-electrostatic wall-fluid inter-action strength h . However, the limits of ∆ upon ap-proaching the filling transition, θ ց θ C , are finite, whichsignals a first-order filling transition (see in particular theinset of Fig. 7(a)). Moreover, for any fixed opening angle θ > θ C , the wedge adsorption ∆ increases with increasingwall charge density σ . Both observations can be under-stood in terms of the strength of the interaction betweenwall and fluid. In case of an increasing wall charge den-sity σ , the increase of ∆ stems from an increase of thecounterion density which is stronger than the accompa-nying decrease of the coion density. This phenomenonis well-known for non-linear Poisson-Boltzmann-like the-ories as the present one. For the case of a decreasingopening angle θ > θ C the growing overlap of the wall-fluid interactions, both the non-electrostatic as well asthe electrostatic one, leads to an increase in the density. Besides these general qualitative trends there are quan-titative differences for the two cases in Fig. 7, which differin the values of the decay length λ . One way to compareFigs. 7(a) and 7(b) is to consider the limits ∆( θ + C ) upon θ ց θ C for a common value of the wall charge density σ . In this case, the shorter-ranged wall-fluid interaction, λ = 1 d (see Fig. 7(a)), leads to higher values of ∆( θ + C )than the longer-ranged one, λ = 2 d (see Fig. 7(b)). How-ever, since shorter decay lengths λ lead to smaller criti-cal opening angles θ C (see Fig. 5), which correspond tostronger overlaps of the wall-fluid interactions of the twowalls of the wedge, an increase in the wedge adsorption ∆is caused mostly for geometrical reasons. Alternatively, ifone compares Fig. 7(a) and 7(b) for a fixed opening angle θ > θ C and a fixed wall charge density σ , the wedge ad-sorption ∆ is larger for the case of the longer-ranged wall-fluid interaction. This can be readily understood giventhe fact that, for fixed opening angle and wall charge, theinteraction strength at a specific point in the system isthe stronger the longer ranged the interaction is.As a second quantity to describe the filling of a wedgethe filling height l w = Γ sym φ (liquid)tot − φ (gas)tot (14)is considered, where Γ sym denotes the excess adsorptionalong the symmetry plane (cell index j = 0) of the wedge:Γ sym := X l ( φ tot; l, − φ (gas)tot ) . (15)The definition of the filling height l w of a wedge is sim-ilar to that of the film thickness l at a planar wall (see ∆ θ (deg) λ = 1d θ − θ C (deg) σ ( un i t s o f e / d ) (a) ∆ θ (deg) λ = 2d σ ( un i t s o f e / d ) (b) FIG. 7. Wedge adsorption ∆ (see Eq. (13)) as function of theopening angle θ of the wedge and of the wall charge density σ . In panel (a) the decay length of the wall-fluid interactionpotential is λ = 1 d, whereas in panel (b) it is λ = 2 d. Similarto the filling height l w shown in Fig. 8, the wedge adsorption∆ increases for increasing wall charge density σ as well as fordecreasing opening angle θ . The limits of ∆ upon approachingthe filling transition, θ ց θ C , are finite, which signals a first-order filling transition. To highlight this, the inset in panel(a) shows a double-logarithmic plot of the wedge adsorption∆ as function of the distance θ − θ C from the filling transition. Eq. (10)). It expresses the distance of the liquid-gas in-terface of the adsorbed film from the tip of the wedge.Figure 8 displays the filling height l w as function of theopening angle θ and of the wall charge σ with the decaylengths λ = 1 d in Fig. 8(a) and λ = 2 d in Fig. 8(b).When discussing the filling height l w one has to accountfor the geometrical effect of an increasing side length l w ( θ ) := d/ sin( θ/
2) of the elementary parallelogramsin the direction of the symmetry plane (see Fig. 1) upondecreasing the opening angle θ . It is equivalent to a fill-ing height of exactly one cell and it is displayed in Fig. 8as a black dashed curve. By comparing the filling height l w ( θ ) with the trend given by the side length l w ( θ ) oneinfers a stronger increase of the former upon approach-ing the filling transition θ ց θ C , which can be attributedto the filling effect. Similar to the wedge adsorption ∆, l w θ (deg) λ = 1d θ − θ C (deg) σ ( un i t s o f e / d ) (a) l w θ (deg) λ = 2d σ ( un i t s o f e / d ) (b) FIG. 8. Filling height at the symmetry plane l w as function ofthe opening angle θ of the wedge and of the wall charge density σ . In panel (a) the decay length of the wall-fluid interactionis λ = 1 d, whereas it is λ = 2 d in panel (b). The dashedblack curve in both panels corresponds to the thickness of thefirst layer of cells on the symmetry axis. The comparison ofthis curve with the curves of the filling height l w shows, thatthe increase of l w close to the critical opening angle θ > ∼ θ C stems from the increasing interactions close to the tip of thewedge. Furthermore the filling height l w increases with bothan increasing wall charge density σ as well as a decreasingopening angle θ . The finite limits for l w upon θ ց θ C pointto a first-order filling transition. Similarly in Fig. 7(a) theinset in panel (a) shows a double-logarithmic plot of the fillingheight l w as function of the distance θ − θ C from the fillingtransition to verify its first-oder nature. the filling height l w increases either upon decreasing theopening angle θ towards the critical opening angle θ C or, for fixed θ > θ C , upon increasing the magnitude ofthe wall charge density σ . The reason for these observedtrends of the filling height l w is again, as for the wedgeadsorption ∆, a consequence of the increased magnitudeof the wall-fluid interaction. Finally, the filling height l w ,as the wedge adsorption, approaches a finite limit upon θ ց θ C , which is in agreement with the expectation of afirst-order filling transition.As shown in Eq. (6), the equilibrium grand potentialΩ eq may contain a contribution scaling proportional to alinear extension L of the system and the corresponding -1-0.8-0.6-0.4-0.200.2 0 60 120 180 β τ d θ (deg) λ = 1d σ ( un i t s o f e / d ) (a) -0.6-0.5-0.4-0.3-0.2-0.100.1 0 60 120 180 β τ d θ (deg) λ = 2d σ ( un i t s o f e / d ) (b) FIG. 9. Line tension τ as function of the opening angle θ andof the wall charge density σ for decay lengths (a) λ = 1 d and(b) λ = 2 d. For small wall charge density σ , the line tension τ is negative for all opening angles θ , whereas for sufficientlylarge θ and σ positive values of τ may occur. coefficient of proportionality of the dimension of an en-ergy per length is called the line tension τ . In the presentcontext of a wedge, the line tension τ measures the struc-tural difference between a wedge and a planar wall, andthe contribution τ L scales with the length L of the wedgealong the z -direction.Figure 9 displays the line tension τ as function of theopening angle θ and of the wall charge density σ for decaylengths λ = 1 d (Fig. 9(a)) and λ = 2 d (Fig. 9(b)). Thequalitative dependence of the line tension τ on the open-ing angle θ turns out to depend on the wall charge den-sity σ : For small wall charge densities the line tension isnegative and it decreases monotonically with decreasingopening angle. For sufficiently large wall charge densitiesthe line tension is positive for large opening angles and, ifthe critical opening angle θ C is small enough, negative forsmall opening angles, i.e., the line tension may dependnon-monotonically on the opening angle. For molecular length scales d ≈ T ≈
300 Kthe order of magnitude of the line tension | τ | ≈ pN is inaccordance with literature [24, 29, 30]. IV. CONCLUSIONS AND SUMMARY
In the present work the filling of charged wedges byelectrolyte solutions has been studied within microscopicclassical density functional theory of a lattice model(Fig. 1). As in previous studies [22–24], considering lat-tice models offers technical advantages over continuummodels, as the former allow for the explicit descriptionof larger parts of the system. The electrolyte solutioncomprises a solvent and a univalent salt. A short-rangedattractive interaction between the fluid particles leads toa liquid-gas phase transition of the bulk electrolyte so-lution (Fig. 2). A fluid-wall interaction derived from aGaussian pair potential (Eq. (4)) gives rise to first-orderwetting transitions of a planar wall in contact with a gasbulk phase (Figs. 3). This first-order wetting transitionof a planar wall can be driven by the wall-fluid interactionstrength or by the surface charge density (Fig. 4). Thecritical opening angle, below which the wedge is filled,depends on the surface charge density and on the decaylength of the wall-fluid interaction (Fig. 5). Upon ap-proaching the critical opening angle from above, a macro-scopically small but microscopically finite amount of fluidis accumulated close to the apex of the wedge (Fig. 6).This observation as well as the finite limits of the wedgeadsorption (Fig. 7), the filling height (Fig. 8), and the linetension (Fig. 9) are compatible with a first-order fillingtransition. Upon increasing the surface charge density,the line tension as function of the opening angle changesfrom a monotonically increasing negative function via afunction exhibiting a positive maximum to a monotoni-cally decreasing positive function (Fig. 9).The unequivocally first-order filling transitions foundwithin the model of the present work are in full agreementwith the general expectation for systems with barriers inthe effective interface potential at the filling transition[17, 18]. Moreover, this is expected to be the case forany electrolyte solution not too close to a critical point,as such systems exhibit barriers in the effective interfacepotential for all conditions of partial wetting [22]. There-fore, the optimistic point of view in Ref. [19] expectingthe experimental accessibility of systems displaying criti-cal filling transitions requires to exclude the vast class ofdilute electrolyte solutions as potential candidates. Onthe other hand, being assured of the first-order nature offilling transitions in the presence of electrolyte solutionsallows one to numerically efficiently set up more realisticmodels, which are not restricted to a lattice for technicalreasons, to quantitatively describe wetting and filling ofcomplex geometries. [1] B. Lin, ed.,
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