Contact Angle Hysteresis and Meniscus Corrugation on Randomly

Article pubs.acs.org/Langmuir

Contact Angle Hysteresis and Meniscus Corrugation on Randomly Heterogeneous Surfaces with Mesa-Type Defects Dimitar Iliev,† Nina Pesheva,‡ and Stanimir Iliev*,‡ †

Fraunhofer Institute for Industrial Mathematics (ITWM), Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. 4, 1113 Sofia, Bulgaria

‡

ABSTRACT: The results of a numerical study of the various characteristics of the static contact of a liquid meniscus with a flat but heterogeneous surface, consisting of two types of homogeneous materials, forming regularly and randomly distributed microscopic defects are presented. The solutions for the meniscus shape are obtained numerically using the full expression of the system free energy functional. The goal is to establish how the magnitude and the limits of the hysteresis interval of the equilibrium contact angle, the Cassie’s angle, and the contact line (CL) roughness exponent are related to the parameters, characterizing the heterogeneous surfacethe equilibrium contact angles on the two materials and their fractions. We compare the results of different ways of determining the averaged contact angle on heterogeneous surfaces. We study the spread of the CL corrugation along the liquid meniscus. We compare our results with the numerical results, obtained using linearized energy functional, and also with experimental results for the CL roughness exponent. The obtained results support the conclusion that some characteristics depends on the type (regular or random) of the heterogeneity pattern.

I. INTRODUCTION Wetting of a liquid on nonideal solid surfaces is still an open problem of general interest.1 The study of this problem is very important, since most of the real surfaces appearing in nature, in the laboratories, and in the different technological processes are not ideal, they are rough and heterogeneous, have finite rigidity, and so forth. As a rule the heterogeneity and the roughness of the solid surface are responsible for the appearance of a set of metastable states with liquid−fluid interfaces which are corrugated in the contact line (CL) neighborhood. Basic integral quantative characteristics of the heterogeneity and/or roughness of the solid surfaces are the magnitude of the hysteresis interval of the equilibrium contact angle (CA) [θr,θa] (where θa is the ″advancing″ and θr is the ″receding″ contact angle), the Cassie’s and Wenzel’s angles, and the magnitude of the roughness exponent of the CL. The equilibrium state of a liquid in contact with solid body with rough and/or heterogeneous surface is given by the classical capillary model.2,3 However, the establishment of relations between the above-mentioned integral characteristics of the solid surface and the specific local characteristics of the heterogeneity and/or roughness such as the equilibrium contact angles on the different homogeneous patches of the surface, the fraction of surface areas of the different materials, the ratio of the true surface area to the total base area, and so forth is still under investigation. In this work we focus on the case when a vertical solid plate, which is flat but with heterogeneous surface, is partially immersed in a tank of liquid. The surface of the plate consists of © 2013 American Chemical Society

two types of nonhysteretic materials, forming microscopic patterns with sharp boundaries. The parameters characterizing the heterogeneous surface are as follows: the equilibrium contact angles on the two types of chemical materials (1) and (2), which we denote by θ(1) and θ(2), and the fractions of the two types of materials, which we denote by f and (1 − f), respectively, (often it is said that one of the materials forms defects on the other4). The Cassie’s angle θc for this type of surface is determined by the equation cos θc = f cos θ (1) + (1 − f )cos θ (2)

(1)

The question, how the integral characteristics of the surface θr, θa, and θc are related to the parameters θ(1) and θ(2), and f, is a subject of both experimental and theoretical studies for many years now. We would like to point out here that the contact angles θ(1) and θ(2), θc, and the defect fraction f, characterizing the heterogeneous surface, are uniquely defined. However, in contrast to this, the angles θr, θa are integral characteristics of the heterogeneous surface, which can be defined and determined in different ways. For the system, considered in the present work, one can determine experimentally the parameters θr, θa by measuring the rise (or depression) height of the meniscus of a liquid against the solid surface,5 using the capillary force (i.e., by the Wilhelmy method)6 or by the Tangent Line Method (TLM).7 The simplest way to find the Received: January 24, 2013 Revised: April 9, 2013 Published: April 15, 2013 5781

dx.doi.org/10.1021/la400328d | Langmuir 2013, 29, 5781−5792

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of the present study. We obtain here solutions for the CL on randomly heterogeneous surface numerically since analytical solutions are out of reach. Previously, solutions for the dynamic CL on randomly heterogeneous surface are obtained within the framework of the phase field model,26−29 and also by Monte Carlo dynamics of three-dimensional solid-on-solid horizontal model.30 Recently, David and Neumann31 obtained solutions within the framework of the capillary theory for the advancing and receding CLs, in which a meniscus forms in contact with a heterogeneous plate, consisting of two homogeneous materials, where square microscopic patches (mesa type defects) of material type (1) are randomly deposited on the second material. In their work, the relations between θr, θa, θc and θ(1), θ(2), f are also studied. The solutions, however, were obtained for the linearized functional of the system free energy, which means that small curvatures of the meniscus are assumed. As was pointed out in ref 31, this represents a sacrifice of accuracy for the sake of faster computation relative to obtaining full solutions. This approach limits the study to the case of weak heterogeneity: |cos θ(1) − cos θ(2)| ≪1. In this work we continue the studies, initiated by David and Neumann, by obtaining full solutions for the meniscus shape for the same randomly heterogeneous plate. We obtain the solutions numerically, and due to that, one does not need to use linearization of the energy functional and also impose the requirement of weak heterogeneity. We briefly reiterate the different definitions for the averaged CAs, used in the literature, and make a comparison of the results, obtained by them. Also, we compare and analyze the differences between the results for periodic defect patterns and random distribution of the defects on the heterogeneous flat solid surface.

macroscopic apparent contact angle (i.e., by the TLM) consists of drawing a line, tangent to the meniscus curve at the top, and measuring the angle of this line. It is important to understand how the different methods of determination of the integral characteristics of the heterogeneous surface affect the results for the CA hysteresis and its relation to the parameters of the heterogeneous surface θ(1), θ(2), θc, and f. The studies of heterogeneous surfaces, where the materials form periodic structures, show that such a relation cannot be established without taking into account details of the patch structure, i.e., the mutual disposition, the shapes and the size of the heterogeneity patterns.8−10 It is an important question whether this continues to hold true also when one has heterogeneous surface with random distribution of patches, when the length scale of the heterogeneity pattern is several orders of magnitude smaller than the capillary length. We are concentrating on this particular case in the present work. A big number of real hysteretic surfaces can be considered as being of this type. The wetting properties of random heterogeneous surfaces, as well as the existence of relations between θr, θa, θc and θ(1), θ(2), f, are studied both in equilibrium and in the quasi-static case.5,11,12 The development of highly precise techniques nowadays allows the creation of different surfaces with controlled disorder. This allows for precise analysis of the relations under consideration. In refs 13−16 the wetting properties were studied of a heterogeneous surface, formed by deposition of small squares (size: 10 × 10 μm2) of chromium, randomly distributed on a glass plate. In the limit of a dilute system of defects and a slightly deformed contact line, the magnitude of the CA hysteresis and the contact line roughness exponent are obtained by the elastic model of the contact line.4 Different formulas are obtained in the case of smooth defects and for mesa-type defectsthe last ones being the subject of study in the present work. For both types of defects one has cos θr − cos θa ∼ f for small fractions f, since in this case the effect of the defects simply adds up. For higher concentrations synergetic effects appear whose influence on the relations between θr, θa, and f is not taken into account. It is of interest to determine what happens in the case of random heterogeneous surfaces when there are no such limitations such as weakness of the heterogeneity and/or that the defects are dilute. For rough superhydrophobic surfaces with periodically distributed pillars with varying pillar size and pitch in the Cassie’s regime a linear and nonlinear variation of the hysteresis in f is experimentally obtained.17,18 However, these results cannot be extrapolated for the case of randomly distributed defects. In the case of smooth defects, Joanny and de Gennes’s elastic model of the CL predicts roughness exponent ζ = 0.5 for l < ld and ζ ≈ 0.3 for l > ld, where ld is the minimum length of contact line over which the CL distorts from its average position by a distance equal to the defect correlation length.19−21 These values of ζ are smaller that the ones obtained experimentally in refs 5, 11, and 13−16. Modifications of the elastic model lead to greater values of ζ up to 0.5 for the dynamic CL.22−24 These results also do not agree with experimental results where usually one has ζ > 0.5. Within the framework of dynamical Monte Carlo algorithms the roughness exponent of driven elastic strings in disordered media for different functional forms of the elastic energy is computed resulting in ζ = 0.63.25 Currently, more precise determination of the value of the roughness exponent in the framework of the classical capillary model is possible and is of interest. This is one of the objectives

II. PROBLEM FORMULATION We are interested here in obtaining the equilibrium states of a liquid phase with free boundary surface in contact with heterogeneous solid wall under the action of the gravitational and capillary forces. We consider the special case of heterogeneity, consisting of homogeneous mesa type defects. This causes the system free energy to be a discontinuous function of the space coordinates. That is why (as we have done in previous studies of similar heterogeneity pattern, i.e., homogeneous mesa defects8,31−33) we resort to employing the integral approach for determining the equilibrium states of the system, i.e., we find the local minimum of the system free energy under condition of constant volume. We consider a liquid of density ρl and volume V with free boundary surface S in an open vessel with flat and rigid vertical walls as shown schematically in Figure 1. We assume that one of the vertical walls is heterogeneous, and the others are homogeneous. We use a reference system in which the heterogeneous wall coincides with the {x = 0}-plane and the z-axis is oriented opposite to the gravitation force g, i.e., z = −g/|g|. The base of the vessel is a domain of rectangular shape D = {(0, x0),(0, y0)}, and the liquid free surface S is, respectively, the set {x, y, u(x, y); x, y ∈ D}. Thus the determination of the equilibrium states of the system reduces to finding the function u(x, y) defined in the domain D. Since we are interested in the liquid meniscus shape in the vicinity of the heterogeneous wall, in the present work we assume that the size x0 of the vessel in x-direction is sufficiently big. Under this assumption, far away from the heterogeneous wall the liquid meniscus is practically horizontal. We note here 5782


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with probability f and with probability (1 − f) with material type (2). In the figures the material of type (1) is shown with dark gray color and the material of type (2) with white color. One such specific realization is shown in Figure 1, where the two types of materials are present with equal fractions, i.e., f = 0.5. We point out here that when the fraction f (or (1 − f)) is small then the parts of material type (1) (or (2)) can be considered as small square defects placed randomly on the background of material type (2) (or (1)). However, when f and (1 − f) are commensurate then the connected parts of materials type (1) and (2) are not squares at all. A characteristic length in the system is the capillary length lc = (γlf/(ρl − ρf)g)1/2, where g is the gravity acceleration constant. It allows one to formulate the problem in dimensionless variables by expressing all the lengths in terms of the capillary length lc. From now on we use the dimensionless variables. For simplicity we keep the same notation, i.e., we use x, y, z, x0, y0, and a instead of x/lc, y/lc, z/lc, x0/lc, y0/lc, and a/lc. In this work we use dimensionless cell size a = 0.0037. This corresponds to a square defect of size 10 μm when the liquid in the vessel is water. The same size of the square defects was used in ref 31. We use a reference system such that the plane z = 0 coincides with the liquid free surface S sufficiently away from x = 0, i.e., one has for x0 ≫ 1

Figure 1. Schematic drawing of the considered system.

that in the study of this system one cannot ignore the gravity even though the contact angles themselves, which are subject of study, do not depend on gravity.33−35 Above the liquid there is a fluid of density ρf and on the boundary surface S between the two phases acts surface tension γ. The free liquid surface S contacts the heterogeneous solid wall Σ (at x = 0) of the vessel forming the contact line which we denote by L. The heterogeneous wall Σ is composed of two types of homogeneous materials (1), (2) on which the liquid forms equilibrium contact angles θ(1) and θ(2) respectively. On the part of the wall denoted by Σ(1) there is a material type (1), and the rest of the wall denoted by Σ(2) is covered by material type (2). Without any limitations to the generality we assume θ(1) < θ(2). Thus the local equilibrium contact angle θ(y, z(0, y)) takes on the following values when the CL is inside the patches

u(x0) = 0

At the contact lines of the liquid free surface S with the side walls of the vessel in the planes {y = 0} and {y = y0} (which are perpendicular to the heterogeneous wall) we require u(x , 0) = u(x , y0 )

∂u(x , y)/dy = 0

≤ θ(y , z(0, y)) ≤ θ

for ∀ x ∈ [0, x0] when y = 0, y0

Thus for the considered system the equilibrium shape of the liquid meniscus is determined by the function u defined in the domain D, satisfying condition 3, either condition 4 or 5, and which minimizes the following functional:

(2a)

θ

(4)

(5)

i.e., if the point {y, z(0, y)} is the inner point of a patch of material (1) or (2). When a point of the CL belongs to the sharp border Σ(1) ∩ Σ(2) between defects with different surface tension, the local equilibrium CA satisfies the double inequality (see ref 33 for more details) (2)

for ∀ x ∈ [0, x0]

This condition is a periodicity condition for the solution and it was used previously also in ref 31. We also obtain solutions for other type of boundary conditions, e.g., for the case when the liquid free surface is orthogonal to the side walls, which are perpendicular to the heterogeneous wall. In this case one requires that

⎧ θ (1) if y , z(0, y) ∈ Σ(1) − (Σ(1) ∩ Σ(2)) ⎪ θ(y , z(0, y)) = ⎨ ⎪ θ (2) if y , z(0, y) ∈ Σ(2) − (Σ(1) ∩ Σ(2)) ⎩

(1)

(3)

J(u) =

∬D (z 2 + +

(2b)

y0

∫0 ∫0

1 + ux2(x , y) + uy2(x , y) + λu) dx dy

z(0, y)

cos θ(y , u(0, y)) dz dy

(6)

where λ is a Lagrange multiplier that determines the liquid volume V. In the first integral of the above expression one takes into account the gravitation force and the surface tension force acting on S and the constant volume condition, respectively. The second integral accounts for the surface tension of the heterogeneous wall Σ. From the condition that the plane z = 0 coincides with liquid free surface S sufficiently away from the heterogeneous wall, it follows that for the considered system λ = 0. Note that David and Neumann31 assume that u2x(x, y) ≪ 1; uy2(x, y) ≪ 1. We just note here that the considered problem of determining the equilibrium meniscus shape has both integral

However, the condition 2b is not a sufficient condition for the equilibrium of the CL at the sharp border of different domains. Equilibrium is reached only when the liquid is situated at the side of the border, corresponding to the domain of better wettability.33 As a consequence of condition 2b, the CL may remain pinned at this point and thus the pinning of the CL is accounted for in the present model. The specific distribution of the heterogeneity pattern is realized in the following way. The vessel wall Σ is represented as a square lattice of unit cells of size a. Every square cell is either covered with material type (1) or (2). Using a random number generator every cell is covered with material type (1) 5783


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solid plate per unit length of the CL L, is γlf cos θ(R⃗ )m⃗ (R⃗ ), where m⃗ (R⃗ ) is a unit vector, normal to the contact line L at R⃗ ∈ L, and pointing outward the liquid−solid interface.37 For an equilibrium plate, the y-component of this force is assumed to be zero. The z-component of the total force per unit width of the plate is thus γlf cos θ(R⃗)m⃗ (R⃗) ·zdL F= ⃗ y0 L (12)

and differential formulations, which are equivalent (for more details, see ref 33). This last condition (λ = 0) simplifies the determination of the numerical solution; however, the requirement that x0 ≫ 1 results in the necessity to obtain numerical solution in a quite large domain. For the considered geometry of the system, the obtaining of the numerical 3D solution in the domain D is not necessary, since as it follows from our previous asymptotic and numerical studies, sufficiently away from the heterogeneous wall, the solution for the meniscus shape S does not depend on y: S ≡ {x, y, u(x)}. The homogeneous solution u(x) ̅ ̅ is obtained analytically in ref 36. For this solution u̅(x), the following relation holds: u ̅ (x ) =

2 − 2 sin φ(x)

∫

A change of variables allows us to write this as γlf y0 F= cos θ(y , h(y)) dy y0 0

∫

(7)

which makes it more evident that the ratio F/γlf obtained from the experiment defines the cosine of the contact angle ⟨cos θ⟩ = F/γlf, spatially averaged over y, i.e., the average angle θ̃ is obtained as the inverse cosine of this quantity.

where φ(x) is the contact angle, which the homogeneous meniscus u(x) makes with a vertical plane set at x. This allows ̅ to simplify significantly the numerical work by obtaining numerical 3D solution only in the domain D1 = {(0, x1), (0, y0)} ⊂ D, where x1 ≪ x0, and requiring that at the boundary x = x1 of the domain D1 the wanted function u coincides with the known analytical solution u̅ in the domain D/D1, i.e., with the known solution for x > x1. The condition for matching the solutions u and u̅ at the line x = x1 u(x1 , y) = u ̅ (x1); y ∈ [0, y0 ] (8)

θ ̃ = cos−1(F /γlf )

ux(x1 , y) = tan−1[sin−1(1 − u 2(x1 , y)/2)]; y ∈ [0, y0 ] (9)

θ ̂ = sin−1(1 − ⟨h(y)⟩2 /2)

for the function u at the line x = x1. Note that at x1 = 0, u̅ is a solution for the equilibrium meniscus shape in contact with the homogeneous wall, forming with it the equilibrium CA θ = φ(0). For convenience, the free liquid surface S = {x, y, u;̅ x ∈ [0,∞), y ∈ [0,y0]} we denote by S̅(θ). In this way the problem of finding the equilibrium meniscus shapes in the domain D is reduced to finding local minimum in the domain D1 of the functional

∬D (z 2 +

(14)

3. The third possibility to define an averaged CA θ̂ is to use the averaged meniscus height. One can obtain experimentally not only the capillary force but the height as well of the meniscus CL. The value of the average rise height ⟨h(y)⟩ defines a meniscus with constant contact line height, corresponding to a homogeneous plate. The CA this meniscus makes with the plate can be interpreted as an average macroscopically observable CA θ̂.11,31 Taking into account eq 7 one gets for θ̂

leads to the following requirement

J(u) =

(13)

(15)

4. The Tangent Line Method leads to a CA θ , which is obtained from the meniscus slope at the point at which the CL has the maximal height: y0

θ = cot−1 ∂u/∂x|0, y ; u(0, ymax ) = max u(0, y) max

y=0

(16)

We are also interested here in the spread, or the pervasion, of the perturbation of the CL along the meniscus profile (i.e., along the x-axis). It can be characterized by the following quantity:

1 + ux2(x , y) + uy2(x , y) ) dx dy

1

+

y0

∫0 ∫0

τ ̅(x) = τ(x)/τ(0)

z(0, y)

cos θ(y , u(0, y)) dz dy

where τ(x) is the difference between the maximal and minimal heights of the meniscus along the y-axis at distance x from the plate

(10)

under conditions 8, 9, and either 4 or 5. After finding this part of the solution, we then extend the solution further in the domain D/D1 with the help of the analytic homogeneous solution. Let us denote by θ(y) the contact angles of the obtained equilibrium meniscus shape u, having contact line L with varying along y height h(y) = u(0, y). We reproduce here different definitions used in the literature for the averaged CA. 1. The simplest way to average the CA is obtained by averaging its local values along the CL L and the averaged CA is denoted by θ̅. θ̅ =

y0

y0

y=0

y=0

τ(x) = max u(x , y) − min u(x , y)

(18)

We calculate the rms roughness w of the CL (defined as in refs 5, 11, 38) as function of the length scale l for the CL configurations w(l) =

∫L θ(L) dL ∫L dL

(17)

σl2(y ̅ ) = (11)

2. Another possible way to define the averaged angle θ̃ is through the capillary force. The capillary force, exerted on the

⟨h(y ̅ )⟩l = 5784

y0 − l /2

1 y0 − l 1 l

∫l/2

y ̅ + l /2

∫y −l/2

1 l

̅

(19)

[h(y) − ⟨h(y ̅ )⟩l ]2 dy

y ̅ + l /2

∫y −l/2

σl2(y ̅ ) dy ̅

h(y) dy

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Figure 2. Different periodic heterogeneous patterns studied (one period is shown in both y- and z-directions): (a) checkerboard patterned surface; (b) vertically striped heterogeneous surface; (c) periodic surface with fraction of the material (1), f = (2k + 1)/(2k + 2).

We study here how well w(l) is fitted by a function Alς, where ζ is the CL roughness exponent, and whether there is a correlation between the contact angle hysteresis and the CL roughness exponent.

vN1, j = vN1− 1, j + Δtan−1(sin−1(1 − vN21− 1, j/2)) j = 0, ..., N2

In this way, one reduces the initial problem to finding a minimum of the function 20, depending on (N1 + 1)(N2 + 1) variables, under conditions 21 or 22 and 23. This problem is solved by standard minimization methods. Here we use the coordinate descent method. This method allows easy parallelization of the numerical algorithm. Taking into account the specificity of the energy functional, the constant step method ε ≪ Δ is used for the variables v0,j, j = 0, ..., N2. The code in (C++) uses the library MPI allowing parallelization. The numerical results are obtained on the IBM supercomputer Blue Gene P using 512 and 1024 processors. The local CA between vi,j and the vertical wall at v0,j is assumed to be the angle between the plane x = 0 and the plane, determined by the points v0,j, v0,j+1, v1,j. The accuracy of the numerical procedure is very high; it allows determination of the contact angle on homogeneous substrates with an error on the order of 0.02° as compared to the theoretical intrinsic contact angle, given by the Young equation.

III. NUMERICAL ALGORITHM We use a standard procedure for finding a numerical solution of the minimization problemthe determination of the meniscus shape. The first step is performing appropriate discretization of the liquid free surface S̅(θ). One can use the methods from refs 39 and 40. However, taking into account the geometry of the system allows one the use of more efficient and time saving methods like the ones in refs 35 and 41. The simplest possible realization35 is used. The domain D1, where the function u(x, y) is defined, is represented as a set of squares of size Δ. In the results obtained below, we use Δ which is 1/20 of the defect size, i.e., we have set Δ = a/20. Respectively, the function u(x, y) we are looking for is represented by the set of values u(xi, yj), defined on the set of points (xi, yj) = (iΔ, jΔ), i = 0, ..., N1, j = 0, ..., N2. The following notation is used: vi , j = u(xi , yj ), f (u , ux , uy) = u 2 +

1 + ux2 + uy2

IV. NUMERICAL RESULTS AND DISCUSSION IV.1. Periodic Mesa-Type Defects. Our main goal is to obtain the values of the integral characteristics θr,θa, and ζ of the heterogeneous solid surface, with which the liquid meniscus makes a contact, and also the propagation of the disturbance of the CL on the meniscus shape (away from the plate) in the case of random distribution of the defects on the plate surface. However, since we would like to see whether and how all these quantities change compared to the case of periodic defects, first, we obtain solutions for the meniscus shape and perform analysis of the solutions in several cases, where the heterogeneity patterns are periodic. We show one period in y- and z-directions in Figure 2 of the studied periodic patterns which are of size (2a × 2a), (2a × a), and (2a × (k + 1)a), respectively. Typical characteristics of the solutions are obtained for the following choice of the equilibrium CAs on the two types of materials: (a) weak heterogeneity (small difference between the values of the CAs of the two materials), θ(1) = 30°, θ(2) = 40°, (these are the values used also in ref 31); (b) strong heterogeneity used in refs 42, 43. In the figures we show the material of type (1) with dark color and the material of type (2) with light color. We begin by obtaining the equilibrium meniscus shapes with the highest and lowest possible (average) heights of the CL. Following ref 31 we call these CLs the receding CL (RCL) and the advancing CL (ACL), respectively. We start the numerical algorithm with approximation of the analytical solutions for the

The energy functional 10 is approximated by35 N1

N2

J ≈ J ̅ = Δ2 ∑ ∑ Fij + cos θ (1)|Σ(1)| + cos θ (2)|Σ(2)| i=0 j=0

(20)

where the following quantities are introduced: Fij = f ((u*)ij , (ux*)ij , (uy*)ij ) (u*)ij = (vi , j + vi + 1, j + vi , j + 1 + vi + 1, j + 1)/4

(ux*)ij = (vi + 1, j + vi + 1, j + 1 − vi , j − vi , j + 1)/(2Δ) (uy*)ij = (vi + 1, j + 1 + vi , j + 1 − vi + 1, j − vi , j)/(2Δ)

By |Σ(1)| and |Σ(2)| we denote the size of the domains Σ(1) and Σ(2). The three-phase CL u(0, y) is approximated with a set of straight lines in the intervals [yj, yj+1] = [jΔ, (j+1)Δ], j = 0, N2 − 1. Quite similar is the approximation of the energy functional if one uses instead the method described in ref 41. The boundary conditions 4, 5, and 9 are approximated, respectively, by vi ,0 = vi , N2

i = 0, ..., N1

vi ,1 = vi ,0 ; vi , N2 = vi , N2 − 1

(21)

i = 0, ..., N1

(23)

(22) 5785


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Figure 3. One period along the y-axes of the numerical solutions for the CL are shown: (a) RCL, s = 1:100; (b) ACL, s = 1:100, for the checkerboard pattern (displayed in Figure 2a), and (c) CL-solution for the vertically striped surface (Figure 2b), s = 1:10.

that since the cell size a ≪ 1, the obtained average CAs for different realizations of the defects differ by less than 0.3°. Case 2(b). It is well-known that for a vertically striped surface the equilibrium solution is unique. Analytic solutions for this case are obtained in ref 42 when the pressure difference in Laplace’s equation of capillarity may be neglected. We show the obtained solution for such surface in Figure 3c for θ(1) = 30°, θ(2) = 40° for the above choice of the parameter values (the scale used here is s = 1:10). We show the obtained values for the different average CAs, θ̅, θ̃, θ̂, defined above, in Table 1 for different combinations of

homogeneous cases S̅(θ(1)) and S̅(θ(2)) in turn. Since the heterogeneous solid surface has periodic structure, we obtain solutions of the minimization problem by imposing periodic boundary conditions in y-direction and find solutions for the meniscus shape for surface domains, which comprise two periods in y for the cases (a) and (b) and (k + 1) periods in the case (c) depicted in Figure 2. The fraction of the two types of materials in the cases (a) and (b) is equal, i.e., f = 0.5, and in case (c) the fraction of the material (1) is f = (2k + 1)/(2k + 2). The pattern (a) is known as a checkerboard type surface and a solution for this case is obtained in ref 44, while for case (b), that of a vertically striped surface solution was obtained previously in refs 32 and 45−47. Case 2(a). The CLs of the obtained periodic solutions of the meniscus shape of a liquid in contact with the heterogeneous chessboard pattern, shown in Figure 2a, in the weak heterogeneity case, θ(1) = 30°, θ(2) = 40°, are displayed in Figure 3apart (one period in y) of the RCLand in Figure 3bpart of the ACL. In these figures, the area of the plate in contact with the liquid (i.e., below the CL) is shown with a darker color. Since τ(0) ≪ a, we use different scales in x- and ydirections in order to show the solution in more detail. If one denotes the ratio of the scales in x- and y-directions by s, we have set s = 1:100 in (a) and (b) parts of Figure 3. As one can see, in both cases parts of the CL lie on the border separating the two types of materials, the material type (2) lying above the CL. This result agrees very well with the theoretical predictions.33 The horizontal parts of the CL-solutions are stuck to the defect borders (i.e., the CL is pinning is observed) where the contact lines of the solutions S̅(θ(1)) and S̅(θ(2)) pass, respectively. The averaged contact angles θ̅ (receding and advancing) (see eq 11) of these solutions are 30.4° (receding) and 39.65° (advancing) correspondingly, and for θ we get the values 30.2° and 40°, respectively. Since the difference between the values of the local angles is less than 3° (for both cases) and the heights of the solutions are very close to the heights of the homogeneous solutions S̅(30°) and S̅(40°), one obtains: θ̅ ≈ θ̃ ≈ θ̂. Specific for this case is that the hysteresis interval [θ̅r, θ̅a] is very close to the interval [θ(1), θ(2)], composed by the equilibrium CA’s for the corresponding homogeneous surfaces. This particularity of the solution holds true not only for θ(1) = 30°, θ(2) = 40°, but also for other values of the CA angles θ(1), θ(2). Since the horizontal parts of the CL are stuck to the defect borders, the obtained solutions depend on the specific realization of the borders between the defects, more precisely on the heights of the set of straight-line segments which form the borders between the defects. We do not study this dependence in detail in the present work; however, we note

Table 1. Different Average CAs in the Case of a Vertically Striped Surface for Different Combinations of the Contact Angles of the Two Materials θ(1)

θ(2)

θ̅ (eq 11)

θ̃ (eq 14)

θ̂ (eq 15)

θc (eq 1)

(θ(1) + θ(2))/2

30° 30° 10° 20° 30° 40°

40° 80° 80° 70° 60° 50°

34.91° 53.73° 44.45° 44.12° 44.3° 44.9°

35.11° 57.51° 52.96° 48.5° 45.9° 45°

34.87° 58.46° 53.31° 49.9° 46.6° 44.93°

35.31° 58.68° 54.6° 50.14° 46.92° 45.22°

35° 55° 45° 45° 45° 45°

the contact angles θ(1), θ(2) of the two materials, forming the heterogeneous surface. In the last two columns of the table we also give Cassie’s angle θc = (cos θ(1) + cos θ(2))/2 and the angle obtained by the simple averaging (θ(1) + θ(2))/2 for all the cases considered. For θ in all cases we get θ = θ(1). The results in Table 1 show that the value of the average CA θ̅ is close to (θ(1) + θ(2))/2, and θ̂ and θ̃ have close values, and they do not differ much from Cassie’s angle θc. Therefore for the considered case (of vertically striped surface) the macroscopic characteristics of the meniscus are well represented by θ̂ and θ̃. The averaged CA θ does not give sufficient information about the characteristics of the meniscus. A similar result is obtained also in ref 33. Case 2(c). When the liquid is in contact with periodic heterogeneous surface of the type shown in Figure 2c, we find that the liquid free surface forms the CLs shown in Figure 4 (where k = 2 is set). In Figure 4a the RCL and in Figure 4b the ACL are given for θ(1) = 30°, θ(2) = 40°. We also show a part (one period of the solution has width 3a) of the obtained solution for the meniscus shape in Figure 4c up to a distance 0.35a from the heterogeneous wall (it has the CL shown in Figure 4a). The numerical studies show that for different values of k the equilibrium state with the RCL again has horizontal parts of the CL stuck to the defect borders where the CL of the homogeneous solution S̅(30°) lies. That is why the averaged 5786


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Figure 4. One period of the numerical solutions for the meniscus is shown in the case of a heterogeneous surface with the pattern in Figure 2c: (a) RCL, s = 1:50; (b) ACL, s = 1:10.5. (c) Part of the numerical solution for the meniscus shape with the RCL shown in (a): only 0.3a of the solution is shown in x-direction in the neighborhood of the heterogeneous wall.

Figure 5. Numerical solutions for a meniscus in contact with a randomly heterogeneous substrate, where the fraction of the material type (1) is f = 0.4: (a) RCL, s = 1:3.5; and (b) ACL, s = 1:3.5; (c) part of the numerical solution for the receding meniscus (corresponding to the CL in (a)); and (d) part of the numerical solution for the advancing meniscus (corresponding to the CL in (b)).

RCA’s for this type of heterogeneous surface are quite close to θ(1), i.e., one has r θ ̅ ≈ θ ̃ ≈ θ ̂ ≈ θ (1) r

r

cos θcs = (cos θ (2) + k cos θ (1))/(1 + k)

For the surface with heterogeneity patterns, shown in Figure 2a and c, the relations between the Cassie’s angle θc, θ̂r, and θ̂a

(24)

a

r

cos θc ≈ (cos θ ̂ + cos θ ̂ )/2

For example, for the above choice of the initial CL and for k = 1, 2, 4, 6, one has θ̅r = 31.61°, 31.22°, 30.77°, 30.65°, respectively. It is worth mentioning here that, in contrast to the case of finding the RCL, in the course of the minimization procedure the CL passes (in vertical direction) through many rows of defects before it reaches the equilibrium ACL. The average height of the CL depends on the value of the parameter kit increases with k and, respectively, the average CA decreases. Thus for k = 1, 2, 4, 6, θ(1) = 30°, and for two values of θ(2), 40° and 70°, we have obtained results for the averaged advancing CAs. These results show that the values of the advancing angles θ̂a and θ̃a are close to θsc a a θ ̂ ≈ θ ̃ ≈ θcs

(26)

(27)

̃r

̃a

and between the Cassie’s angle, θ , and θ a

r

cos θc ≈ (cos θ ̃ + cos θ ̃ )/2

(28)

are satisfied with good precision. In the first case, shown in Figure 2a, it follows from ⟨θ(y)⟩r ≈ θ(1), ⟨θ(y)⟩a ≈ θ(2), and f = 0.5, and in the second, shown in Figure 2c, from eq 24, eq 25, and f = (2k + 1)/(2k + 2). The analysis of the numerical solutions obtained for the equilibrium meniscus shapes, in the weak heterogeneity case θ(1) = 30°, θ(2) = 40°, shows that the perturbation of the CL, caused by the heterogeneity, disturbs the meniscus shape at distances 500 μm and discussed there. In the results presented here this happens for smaller values of l, which can be explained by the small domain of the solution and the use of periodic boundary conditions. Taking the above into account, in order to eliminate (or reduce) the effect of the periodic boundary conditions we fit w for small values of l. The performed numerical analysis shows that for the RCL and the ACL in the interval [0.5−80 μm] the roughness exponent ζ exhibits a range of values [0.57− 0.67] (and in the interval [25−80 μm] ζ ∈ [0.5, 0.63]). If one decreases the upper bound of the fitting interval, ζ increases and on the interval [0.5−40 μm] one has ζ ∈ [0.67, 0.75]. The general tendency is that for small concentrations f (respectively

conditions has bigger variation of the height, i.e., it covers more rows of defect cells (in the vertical direction) than the solutions obtained under periodic boundary conditions. The left and the right ends of the CL are on the lowest and the highest of these rows, respectively. A similar effect of “inclination” of the CL is observed experimentally in refs 5, 16. Comparison of the CLs obtained under periodic boundary conditions and different number of defects (Figure 5a and Figure 6b) reveals that the higher number of defects also leads to an increase of the height variation of the CL. In the case of weak heterogeneity and periodic boundary conditions a good agreement is found with the results for θ̂, obtained from the full model and the linearized version, as can be expected. This supports the conclusion, made in ref 31, for a good approximation of eq 27. The analysis of the attenuation of the perturbations of the CL, induced by the heterogeneity of the solid surface on the equilibrium meniscus, shows that for periodic boundary conditions one has τ(x) ∼ 0.01 at distances x from the wall ̅ in the interval (16a, 24a). However, if orthogonal boundary conditions are imposed, τ(x) increases, e.g., at a distance x ∼ ̅ 25a it is still on the order of ∼0.4. While for the periodic heterogeneities), presented in section 4.1, the perturbations (caused by similar defect sizes) spread along the meniscus surface are approximately 1 defect, now even for a weak heterogeneity the pervasion of the perturbations is a magnitude bigger. Macro-perturbations of the CLs (of size up to 40 defects) pervade deep inside the liquid interface, far away from the contact with the vertical heterogeneous wall (see also Figure 5c,d). In conclusion, densely distributed random defects can produce large scale deformations of the meniscus. The appearance of collective effects is also observed in the type of the function w(l) (the rms roughness of the CL (eq 19). We show in Figure 7a the obtained functions w(l) for the presented CLs in Figures 5 and 6. For the sake of easier comparison with the previous results in refs 5, 11, and 31 the presented results are in dimensional units of a water meniscus. For small values of l, w(l) is fitted well with a power function Alζ. For l in the interval [0.5−100 μm], the roughness exponent ζ for the different CLs varies in the interval [0.52−0.7]. If one decreases the upper bound of the fitting interval, the roughness exponent ζ increases and in the interval [0.5−20 μm] one gets ζ = 0.85 for all CLs. 5789


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Figure 9. (a) Dependence of the difference (cos θri − cos θai ) (which is also used as a measure of the CA hysteresis) on the defect fraction f is shown for θ(1) = 30°, θ(2) = 80°. The symbols are the numerical results and the subscript i denotes the different ways in which the averaging is performed: circles, θ1 = θ̅; triangles, θ2 = θ̃; and squares, θ3 = θ̂. The dashed line is the fitting function g( f) = 1.4f(1 − f). (b) Results for G = Δ(cos θ1)/( f(1 − f)) as a function of the difference cos θ(1)− cos θ(2) are presented at fixed θ(1) = 30°. The squares correspond to the following set of values for θ(2) = 40°, 50°, ..., 90°. The dashed line is the linear fit Q1; the quadratic fit Q2 is shown with a dotted line.

meniscus τ(20a) = 0.01 for θ(2) = 40°, and it increases ̅ smoothly to 0.054 for θ(2) = 90°; and for the advancing meniscus τ(20a) = 0.035 for θ(2) = 40° and it increases to 0.079 ̅ for θ(2) = 90°. Our results, obtained above for θ(2) = 80°, θ(1) = 30°, and different fractions f, for the advancing and receding CAs, show that half of their sum, which we denote by δ cos θi, where the subscript i is used here to denote the different ways in which the averaged CAs (θ̅, θ̃, θ̂) are obtained, is close to Cassie’s angle, calculated for the corresponding value of the fraction f. That is, one has

(1 − f)) up to 0.5, the value of ζ is smaller (in the left part of the interval) and with increasing f (respectively (1 − f)) up to 0.5 the value of ζ increases (and it is in the right part of the interval). The results obtained here for the roughness exponent ζ agree well with the experimental values found in Decker and Garoff5 for water on a fresh Aquapel sample ζ ∈ [0.66, 0.71]. The value of ζ = 0.51 found experimentally in refs 14 and 16 for l > 25 μm in the case of a water meniscus in contact with a plate with randomly distributed square defects of size 10 μm (the same type of defects and the same size as the ones considered in the present study) with concentration 22% (when the plate is in slow motion) is close to the value ζ = 0.5, obtained here for concentration f = 0.2 (see Figure 8a), when fitting w(l) in the interval [25 μm, 80 μm]. Our results also support the increase of the exponent ζ obtained in these references when fitting w(l) for l < 25 μm. Next, we study how the function w(l) behaves for different values of θ(2) at fixed concentration f = 0.4 and for fixed value of θ(1) = 30°. These results are shown in Figure 8b. One can see that at fixed l with increasing the difference between the angles θ(2) − θ(1) the value of w(l) also increases. Comparison of the results from Figure 8a,b for w(l) at small values of l reveal that one gets the same function w(l) for different values of the parameters from both figures. For example, the function w(l) is the same for θ(1) = 30°, θ(2) = 50°, and f = 0.4 (from Figure 8b) and for θ(1) = 30°, θ(2) = 80°, and f = 0.8 (from Figure 8a). However, the hysteresis intervals in these two cases are different. For the first set of parameters the hysteresis interval is [39.1°, 45.14°], while for the second one has [34°, 52.3°]. We interpret this result that there is no correlation between the contact line roughness and the contact angle hysteresis. This conclusion also corroborates the experimental results.5 The increase of the difference θ(2) − θ(1) also leads, as can be expected, to an increase of the distance on the liquid surface to the heterogeneous wall, where the effect of heterogeneity is discernible. Thus, we find the following solutions for the menisci for fixed θ(1) = 30° and f = 0.4: one has for a receding

δ cos θi ≡

cos θir + cos θia ≈ cos θc 2

(29)

for the three different types of averaging of the CAs (θ1 = θ̅, θ2 = θ̃, θ3 = θ̂). In Figure 9a the results for the difference between the cosines of the averaged receding and averaged advancing CAs Δ(cos θi) = cos θir − cos θia

as a function of the defect fraction f are presented for the three types of averaging: θ1 = θ̅, solid triangles, using the CA, averaged over the CL; θ2 = θ̃, solid circles, using the averaged cosine of the CA through the capillary force; θ3 = θ̂, solid squares, using the averaged CL height. The results for Δ(cos θ2) and Δ(cos θ3) are fitted by the function g(f) = 1.4f(1 − f), shown with dashed line. The results for Δ(cos θ3) for the weak heterogeneity case θ(1) = 30°, θ(2) = 40°, presented in Table 3 and in Figure 12 in ref 31, are also fitted well by a quadratic function g( f) with a different coefficient, resulting in g( f) = 0.9f(1 − f). Our results for the same case corroborate this result. It seems reasonable to suppose that the coefficient of the quadratic function depends on the difference cos θ(1)− cos θ(2), i.e., one can write Δ(cos θi) = G(cos θ (1) − cos θ (2))f (1 − f ) i = 2, 3 5790

(30) dx.doi.org/10.1021/la400328d | Langmuir 2013, 29, 5781−5792

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David and Neumann31 (who are using the averaged CL height for the determination of the averaged CA in their work) suppose that the function Δ (respectively the coefficient G) depends linearly on the difference cos θ(1) − cos θ(2). In order to analyze the behavior of the function G(cos θ(1) − cos θ(2)) on its argument, we analyze the obtained solutions for the equilibrium receding and advancing menisci for fixed values of θ(1) = 30° and concentration f = 0.4 and for different values of θ(2). From eq 30 for fixed θ(1) and f, one can determine G(cos θ(1) − cos θ(2)) as function of Δ(cos θi), i = 2, 3.

Here, solutions for the liquid meniscus shape in contact with more realistic heterogeneous surfaces, consisting of two types of homogeneous materials, forming randomly distributed microscopic defects, are obtained numerically for the first time using the full expression of the system free energy functional. It is shown that unlike the case of regularly patterned heterogeneous surfaces, for randomly heterogeneous surfaces with microscopic patterns the hysteresis interval [θr, θa] is uniquely determined by the equilibrium CAs θ(1), θ(2) of the two types of materials and the defect fraction f. It is demonstrated that using different ways of defining the averaged CA’s (i.e., using the average CL height, the capillary force, and the simple averaging along the CL) one gets results which are close, while for the regularly patterned surfaces different scenarios are possible. It seems preferable to use the averaged CAs, determined through the averaged cosine and the averaged CL height. It is confirmed that the half of the sum of the cosines of the receding and advancing CAs θr, θa equals the cosine of Cassie’s angle. For randomly heterogeneous surfaces, in contrast to the previous hypothesis about a linear dependence on the concentration of defects f for the difference of the cosines of the receding and advancing CAs θr, θa, we obtain a quadratic function in f (where the coefficient also depends on the difference of the cosines of θr, θa). This function seems more realistic since from this dependence one naturally obtains that the contact angle hysteresis vanishes, when f goes to zero and respectively to one, as it should be. For randomly heterogeneous surfaces we find that the microdefects cause a corrugation of the CL which pervades deep inside the meniscus up to distances which are several tens of times larger than the defect characteristic size. We find that the contact line roughness increases with the increase of the CA θ(2) (while keeping θ(1) fixed) with defect concentration f. It is demonstrated why, as the experiments show, there is no correlation between the contact line roughness and the contact angle hysteresis. For randomly heterogeneous surfaces with microdefects, the full model leads to a CL roughness exponent, whose value agrees well with its experimentally determined values.

G(cos θ (1) − cos θ (2)) = Δ(cos θi)/(f (1 − f )) i = 2, 3

(31)

The results for G as function of the difference cos θ − cos θ(2) for the following set of values for θ(2) = 40°, 50°, ..., 90° are shown in Figure 9b with squares. We have added to the results also the point (0, 0) since when θ(2) → θ(1), Δ(cos θi) → 0, i = 2, 3. As can be seen from Figure 9b, in the interval 40°−90° the dependence is linear. The dashed line in Figure 9b shows the linear function Q1(cos θ(1) − cos θ(2)) (1)

Q 1(cos θ (1) − cos θ (2)) = cos 40° − cos 30° + 2(cos θ (2) − cos θ (1))

(32)

Fitting the results for G by a quadratic function (denoted here by Q2) which also passes through the point (0, 0) is shown in Figure 9b with dotted line. This function has the following form Q 2 ≡ (cos θ (1) − cos θ (2))(1.1 + cos θ (1) − cos θ (2)) (33)

The relations 29 and 30 allow one to express the averaged receding and advancing CAs θri , θai , i = 2, 3 (i.e., when i = 2, defined using the averaged cosine of the CA and when i = 3, defined using the averaged CL height) through the surface parameters θ(1), θ(2), and f

■

θir = cos−1(cos θc + Gf (1 − f )/2) θia

−1

= cos (cos θc − Gf (1 − f )/2)

i = 2, 3

AUTHOR INFORMATION

Corresponding Author

(34)

*E-mail: [email protected]. Notes

V. SUMMARY AND CONCLUSION In this work we focused on the study of a flat but heterogeneous surface, consisting of two types of materials, forming microscopic nonhysteretic patterns, the size of ∼1/270 of the capillary length with sharp boundaries, i.e., mesa type defects. We studied how the basic integral quantative characteristics of the heterogeneous surfaces, as the magnitude of the hysteresis interval [θr, θa] of the equilibrium CA, the Cassie’s angle θc, and the magnitude of the roughness exponent of the CL, are related to the parameters, characterizing the heterogeneous surface: the equilibrium contact angles θ(1) and θ(2) on the two types of chemical materials, and their respective fractions f and (1 − f). Both periodic and randomly heterogeneous surfaces are studied in order to see whether there is a difference in the behavior of the integral characteristics of the heterogeneous surfaces depending on the type of heterogeneity pattern. Four different ways for obtaining the averaged CA were compared. We also studied the pervasion of the CL perturbation along the fluid meniscus.

The authors declare no competing financial interest.

■

REFERENCES

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Contact Angle Hysteresis and Meniscus Corrugation on Randomly

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