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The present study investigates moving contact lines in microfluidic confinements with rough topographies modeled with random generating functions...
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Combined Effects of Surface Roughness and Wetting Characteristics on the Moving Contact Line in Microchannel Flows Debapriya Chakraborty, Naga Neehar Dingari, and Suman Chakraborty* Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur-721302, India S Supporting Information *

ABSTRACT: The present study investigates moving contact lines in microfluidic confinements with rough topographies modeled with random generating functions. Using matched asymptotic expansion, the description of the whole contact line is obtained and the dynamic contact angle is extracted by extrapolating the bulk meniscus to the channel wall. Significant variations are observed in the contact angle because of the heterogeneities of the confining walls of the microfluidic channel. The effects of the surface wetting condition also play a crucial role in altering the description of the contact line bearing particular nontrivial interactions with the topological features of the solid boundaries. In an effort to assess the underlying consequences, two different surface wetting conditions are studied; namely, complete wetting substrate and partial wetting substrate. Our studies reveal that the consequent wetting characteristics are strongly influenced by action of intermolecular forces in presence of surface roughness. The effect of slip, correlation length, and roughness parameters on the dynamic contact angle have been also investigated.

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community primarily through mesoscopic formalisms6,7 or with full-scale molecular dynamics simulations.8−11 Comprehensive efforts have been devoted in the literature to analyze the contact line from a continuum perspective.12 The contact line, in essence, may be described by decomposing the contact region into various subdomains considering the relative dominance of various forces over different length scales.3,13,14 For example, close to the substrate viscous forces, surface tension forces and intermolecular forces govern the evolution of a thin film, whereas deeper in the bulk the substrate−fluid intermolecular forces cease to be important. However, it is difficult to simulate the entire contact line inside a narrow fluidic confinement from the thin film region (close to the wall) to a resolvable meniscus because of the fact that the near-wall and the far-wall regions may differ in characteristic length scales by several orders. To resolve this issue, one may obtain the dynamic evolution of the interface through an asymptotic matching of two regions,13 namely, an outer region far from the contact line and an inner region formed by a thin lubricating film adhering to the channel walls. Different kinds of inner regions may be formed on the basis of wetting behavior of the films. In the purely wetting limit, the inner region may be described by the evolution of the precursor film,14 whereas for partially wetting surfaces the same may be described by a wedge shaped region with slip at the contact line.15 An effective or apparent contact angle may be defined as an extrapolation of the outer profile of the interface toward the solid substrate.14 Because this description of the contact angle depends on the

INTRODUCTION The physics of interfacial interactions in narrow fluidic confinements has given rise to many seemingly unresolved anomalies as attributed to a complex interplay between interfacial phenomena over small scales and topographical features of the confining system boundaries. Dynamical evolution of capillary fronts in microchannels and nanochannels offers with one such complicated scenario, which essentially stems from the nonintegrable stress1,2 arising at the three phase contact line. Interestingly, the linearity in the stress−strain relationship for a nondeformable solid with no-slip conditions at the three phase contact line would ensure infinite viscous dissipation,2 which is physically impossible. Not only that, the physical fact that a capillary front moves over a solid substrate despite no-slip boundary conditions at the fluid−solid interface had also appeared to be a mathematical paradox for a long time. To resolve such apparently anomalous trends, researchers in many cases had abstracted the underlying details of interfacial interactions with the notion of slip-based hydrodynamic boundary conditions3,4 or with a thin film, which typically runs ahead of the meniscus in the form of precursor films.5 In addition to the critical aspects of interfacial interactions mentioned above, the effects of surface roughness on the resultant contact line dynamics remain far from being trivial. This may be attributed to the fact that, irrespective of the nature of wetting from a global perspective, the local contact line dynamics is significantly dictated by small-scale variations in the surface roughness characteristics, which cannot be merely described solely as a function of the averaged roughness heights of the surface elements. Such issues are further complicated by the existence of disparate physical scales governing the problem, which have been addressed by the research © 2012 American Chemical Society

Received: September 7, 2012 Revised: November 5, 2012 Published: November 6, 2012 16701

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like behavior of the apparent contact line. The substrate variations are captured through the geometric constraints through this apparent contact angle imposed by the substrate. Further, Savva et al.53 model the substrate as families of random functions by extending their earlier deterministic model50 through an effective linearization for small substrate amplitudes, to study the droplet shift and contact line fluctuation for slowly varying topographies. They also perform stability analysis to identify the regimes of stable and unstable equilibriums. One of the interesting conclusions drawn from their work51 is that the substrate roughness inhibits wetting. However, instead of considering a free droplet, the present study analyzes the contact line dynamics in a confined microchannel, which leads to the observation that the combined influence of substrate wetting characteristics and surface roughness, in a microconfined fluidic environment, may either enhance or inhibit wetting of a continuous capillary in a nontrivial manner, depending on the nature of wetting of the substrate and the topographical parameters of the rough surface, as demonstrated subsequently.

contact line velocity, the contact angle also varies dynamically as a meniscus evolves. In many descriptions of contact line dynamics outlined as above, the substrate is traditionally modeled either as an idealized flat surface or a rough surface with a representative roughness length scale disregarding its topographical details. In reality, however, substrates are likely to contain heterogeneous topographical features. Studies on the effect of surface roughness on wetting phenomena,16−28 however, are not new. In fact, several theoretical models have been developed to describe the dynamics of moving contact line over rough surfaces.22,29−34 Recent studies have concluded that roughness tends to lower the surface energy when the fluid has favorable interaction with the surface, whereas it may also enhance the surface energy when the interaction energies are loose as confirmed by molecular dynamics simulations of the LenardJones liquid for both smooth and rough surfaces.35,36 Importantly, all the above-mentioned studies considered wetting over open surface but did not consider the effects of a confined environment. The confinement effect, however, may turn out to be particularly interesting, in purview of the fact that, for such conduits, the roughness length scales may turn out to be comparable with the characteristic system length scales. Notwithstanding the lack of theoretical treatment highlighting this specific issue, nevertheless, several experimental studies have been conducted to understand the dynamics of the contact angle over rough surfaces.37−43 Further, several studies have been reported recently in the literature on the effects of roughness and hydrodynamic boundary conditions,44 slippage and contact angle,45 and wetting on patterned surfaces.46,47 Interested readers may refer to the recent reviews in this subject.48,49 Here, we address the moving contact line over a rough surface with random inhomogeneities, from a theoretical standpoint, considering a microconfined environment. Local dynamics of such flows are complicated and are subjected to small-scale variations in the topography of the surface. These topographical features play a significant role in modifying the apparent contact angles near the three phase contact line. Accordingly, we aim to describe this effect of a moving contact line in a narrow confinement with randomly generated rough surfaces as a function of the pertinent topographical details. In essence, we envisage in predicting dynamic contact angle variations over such rough surfaces based on random surface profiles through a matched asymptotic analysis by conceptualizing the net effect in terms of an apparent contact angle over a hypothetical flat surface. The apparent dynamic contact angle acts as an effective boundary condition that allows one to abstract the topographical details of the substrate involved by considering an equivalent flat surface, where the description of the dynamic contact angle itself takes the sole burden of representing the geometrical and statistical properties of real corrugated surface without compromising on the essential physics of the problem. It is also important to mention in the context of the objective of the present work that motion of the contact line of an isolated droplet over a topographical substrate has been considered in few recent studies as well.50−53 In Savva and Kalliadasis,50 the flow in vicinity of the contact line of the droplet front is matched asymptotically with the flow at the bulk, using a singular perturbation approach, considering deterministic substrate topography. Their study demonstrates the existence of multiple equilibrium states causing hysteresis

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PROBLEM DESCRIPTION We consider an advancing gas−liquid interface moving over rough surfaces of a narrow confinement of width 2b as shown in Figure 1. The cross-sectional averaged velocity of the

Figure 1. Schematic of a microchannel with meniscus, showing inner and outer regions. Inner region for two different wetting surface − I. Complete wetting; II. Partial wetting, are depicted in the boxes shown in the right. O() refers to the order of magnitude.

capillary front is taken as U. The meniscus formed at the interface of the two phases (liquid−gas system) is decomposed into two regions, namely, inner and outer regions.13,14 The outer region consists of a globally resolvable meniscus and the inner region describes a region in the vicinity of the wall. If the surface is purely wetting, the inner region extends in the form of a precursor film over the rough surfaces of the confinement. For partially wetting substrates, the precursor film ceases to exist and the inner region describes only a small wedgelike film. We consider the surfaces of the confinement to contain random surface inhomogeneities. We describe the surface topography in terms of the local axial coordinate (x) as ξ = ξ(x). The surface roughness (ξ) is characterized by two parameters, namely, ξrms and Rcorr, which represent the rootmean-square (RMS) height and correlation length, respectively. The correlation function is an indicator of the repetitiveness of the surface roughness profile. Different correlation functions may give rise to different wetting characteristics of the valleys created by the concerned surface roughness elements. The random roughness profiles are generated here following a procedure similar to that outlined in Garcia and Stoll.54 In essence, the roughness function is generated by a large number of uniformly distributed random variables in the interval [0, 1], which are added to obtain an uncorrelated distribution ξu(x). It 16702

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yields an expectation of ⟨ξu(x)⟩ = 0 and correlation⟨ξu(x)ξu(y)⟩ = δxy. Using the Central Limit theorem, this distribution may be considered as Gaussian, which may be convoluted with Gaussian (or exponential) distribution to obtain a Gaussian (or exponentially) correlated distribution ξ(x) given by: ξ(x) = L1/2π −1/4

∞

⎛

using molecular simulations.56 The advancement of the tip of the film is often described by the spreading coefficient, as given by: S = σsg − σsl σlg. In the limit of high spreading coefficient as considered in this case, the tip of the precursor film may be considered far away from the meniscus, and it can be decoupled from the advancing meniscus (Figure 2).

2⎞

∫−∞ exp⎜⎝− (x −2L2x′)

⎟ξu(x′)dx′ ⎠

(1)

The above expression satisfies the expectation ⟨ξ(x)⟩ = 0 and the correlation function ⟨ξ(x)ξ(y)⟩ = exp(−((x − y)2/4L2)). Further, the correlation length and the RMS height for such a distribution may be expressed as: 2 R corr =

∫ x 2⟨ξ(0)ξ(x)⟩dx/∫ ⟨ξ(0)ξ(x)⟩dx = 2L2

(2a)

2 ξRMS =

∫ ξ 2(x)dx/L

(2b)

Figure 2. Schematic of a meniscus and scaling in (axial, transverse) directions for inner and outer regions of completely wetting substrate. b is the half width of the microchannel and O() refers to the order of magnitude.

It may be noted that Fourier Transform of 2a is equal to the product of Fourier transforms of Gaussian distribution and ξ(x) (by the property of convolution). Hence, the integral would simply imply the inverse Fourier transform of the product of these two Fourier transforms.

The hydrodynamics of the thin film may be described by neglecting the inertia terms in comparison to the viscous terms in the Stokes flow regime, so that the appropriate form of the momentum equation is given by:

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μ

MATHEMETICAL FORMULATION Outer Region. The outer region is a region far away from the three phase contact line, where the normal stress is balanced by the surface tension only and is mathematically given in nondimensional form (pressures are nondimensionalised by σlg/b and the coordinates are nondimenionalised by the half width of the channel b): ⎞ ⎛ rxx 1 ⎟⎟ − P − Pg = ⎜⎜ 2 3/2 r(1 + rx2)1/2 ⎠ ⎝ (1 + rx )

where P and Pg is the liquid and gas pressures respectively, σlg is the liquid−gas surface tension coefficient, (x,r) represents the coordinates of the liquid−gas meniscus and the subscript denotes the derivative of with respect to a variable. Fixing the origin of the outer region coordinate system at the channel centerline, we obtain: (4a)

rx(x → 0+) → +∞(centerline symmetry)

(4b)

(5)

where μ is the dynamic viscosity of the fluid (in the present analysis, the viscosity of the gas phase is neglected as compared to that of the liquid phase), u is the axial velocity, P is the liquid pressure, and (x,y) are the local coordinates. The pressure gradient accounting for the intermolecular force (approximated using Derjaguin approximation57,58) and surface tension (curvature approximated in lubrication limit) may be expressed as (Supporting Information):

(3)

r(0) = 0

∂ 2u dP = dx ∂y 2

−

⎞ dP A ∂ ⎛ ∂ 2h = ⎜σlg 2 − 3⎟ dx ∂x ⎝ ∂x 6π (h − ξ) ⎠

(6)

Substituting eq 6 into eq 5, and solving it using no-slip boundary condition u(z = ξ(x)) = 0 and no shear at the free surface ∂u/∂z(z = h(x)) = 0 with further noting the statement of conservation of mass as: −dQ/dx = ∂h/∂t (where Q is the volume flow rate and t is the time), the equation for thin film incorporating the effects of roughness as well as the disjoining pressure in a simplified 2D framework is given by: ⎞⎞ ⎛ ∂ 2h 1 ∂ ⎛ A ∂h 3 ∂ + ⎟⎟⎟ = 0 ⎜σlg 2 − ⎜⎜(h − ξ) 3μ ∂x ⎝ ∂t ∂x ⎝ ∂x 6π (h − ξ)3 ⎠⎠

It may be noted that the remaining boundary condition for nondimensionalised eq 3 cannot be directly specified because of the fact that, as r → 1, the intermolecular forces of interaction between the fluid and the solid substrate tend to play significant roles, which are specified through the formulation of the inner region. We asymptotically match these two solutions to obtain the complete solution. Further, it is important to note that the outer solution remains the same for both the cases of completely wetting and partially wetting substrates, but the inner region description is different for these two cases and will be independently described in the subsequent subsections. The effect of the confining boundary of the microchannel is explicitly manifested through eq 3. The outer region feels the effect of the wall only through an implicit coupling of the two regions as dictated by their matching boundary conditions. Inner Region: Complete Wetting Case. For a completely wetting surface, a thin film called precursor film remains present which has been verified both experimentally55 and

(7)

In the limiting case corresponding to a flat smooth surface ξ(x) =0, we obtain the same equation governing film thickness as derived by Kalliadasis and Chang.14 In effect, we have extended the analysis for the completely wetting regime as obtained by Kalliadasis and Chang14 to include the effects of surface roughness. We introduce Galilean transformation (x → x − Ut) for a moving coordinate transformation with the meniscus moving at an average velocity U.59 The axial and the lateral length scales60 in the inner region being related to Ca1/3 and Ca2/3 respectively (where Ca = (μU/σlg) is the capillary number), the coordinates may be normalized as: X = (x − x0/bCa1/3) and H = (h/bCa2/3) where x0 (>0) denotes the origin of the coordinate system (X, H). As per the sign convention adopted in this work, the 16703

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Hamaker’s constant is negative for a wetting fluid (i.e., A = −| A|). We define a nondimensional molecular length scale as υ = ((|A|/6πσlgb2))1/2, which depicts the effects of the intermolecular force. We normalize the roughness parameter using the RMS value of the roughness profile as ζ = ξ/ξrms. Accordingly, we express the governing equation of thin films in a dimensionless form to yield: −3

⎞⎞ ∂H ∂ ⎛ ∂ ⎛ ∂ 2H ε ⎜⎜(H − ε1ζ )3 ⎜ 2 + ⎟⎟⎟ = 0 + ∂X ∂X ⎝ ∂X ⎝ ∂X (H − ε1ζ )3 ⎠⎠ (8)

Figure 3. Schematic of a meniscus and scaling in (axial, transverse) directions for inner, intermediate and outer regions for partially wetting substrate. b is the half width of the microchannel, Ca is the capillary number, λ is the slip length, and O() refers to the order of magnitude.

where ε = (υ/Ca)2 represents the scaled molecular length depicting the influence of the intermolecular force on the contact line. In eq 8, the roughness parameter ε1 = (ξrms/ bCa2/3) represents the scaled RMS roughness relative to the characteristic channel dimension. If the roughness height is greater than that of the order of the Bretherton film thickness scale60 by several orders (ξrms ≫ O(bCa2/3)), it would essentially modify the scaling of the inner region by the roughness length scale rather than the film thickness scale. For large film thickness (h ≫ ξrms) in comparison to the roughness height, the surface asperities may be replaced by an effective slip on the surface.61 When the roughness height is comparable to the film thickness (ξrms ∼ O(bCa2/3) ∼ Rcorr), the film thickness as well as the roughness heights are both scaled by bCa2/3, as considered in the present case, with the validity of the lubrication approximations.62 Using the fact that in the asymptotic limit of X → ∞, the tip of the precursor film merges with the wall (H → ε1ζ), and integrating eq 8 once, we get: 3(H − ε1ζ )2 = ((H − ε1ζ )4 HXXX − 3ε(H − ε1ζ )x )

u(z = ξ(x)) + U = λ 2 − α(h − ξ)α − 1

z = ξ(x)

∂u (z = h(x)) = 0 ∂z

(11a)

(11b)

where U is the average velocity of the meniscus and α is a parameter that assumes a value of unity corresponding to the usual Navier’s slip model. Deriving the lubrication equation for the thin film in this region similar to the precursor film model, that is by applying Galilean transformation (in a quasi-steady sense) followed by integrating once, one may obtain: ⎞ ⎛ ∂ 3h ⎞⎛ (h − ξ)2 + (h − ξ)α λ 2 − α⎟ = Ca ⎜ 3 ⎟⎜ 3 ⎝ ∂x ⎠⎝ ⎠

(9)

(12)

We nondimensionalise the axial and lateral length scales as: l = (xCa1/3/31/(2−α)) and η = (h/31/(2−α)λ). Nondimensionalizing the roughness parameter as: ζ = ξ/ξrms, eq 12 takes the form:

The asymptotic matching of eq 9 with that of the outer region equation (eq 3) results in the asymptotic limit of X → −∞, the value of the height of the scaled interface tends to as quadratic form of H = (X2/2 + C3), and the pressure as P2/2 = −C3 (Supporting Information). The parameter C3 represents the correction factor to the curvature and implicitly dictates the influence of the inner region over the outer solution and consequently determines the apparent dynamic contact angle as: θdyn = tan−1( −Ca1/3 −2C3 )

∂u ∂z

⎛ ∂ 3η ⎞ ⎜ 3 ⎟((η − ε2ζ )2 + (η − ε2ζ )α ) = 3 ⎝ ∂l ⎠

(13)

where ε2 = (ξrms/31/(2−α)λ) is a parameter depicting the roughness length scale relative to the slip length. It needs to be noted that, in the completely wetting region, we define the roughness parameter ε1 as the ratio of the roughness length scale to that of the capillary length scale, whereas for the partially wetting region the RMS roughness is scaled with the slip length and expressed in terms of (ε2). This is a distinctive feature of the scales of the inner region, which governs the physics of the contact line depending on the type of the wetting substrate. Further, the inner region cannot be asymptotically matched directly to the outer region solution as was done in the previous case of the precursor region, owing to the extent of the lateral length scale. Hence, we introduce an intermediate region, which matches inner and outer regions in their respective limiting cases (Figure 3). Physically, this intermediate region is also characterized by the balance of surface tension and viscous forces and may be represented by the asymptotic description of the inner region. It may also be represented in the dimensional form same as eq 12. The curvature of the intermediate region needs to match asymptotically to a parabolic profile of the outer region solution.

(10)

Inner Region: Partial Wetting. Low values of the spreading coefficient S often result in a partial wetting surface, when the tip of the precursor film cannot be considered located at infinite distance from the moving meniscus, or in other words the solution of the thin film region may not be treated decoupled as was considered for high spreading coefficient. To relieve the stress singularity at the three phase contact line for this case, we introduce a slip length λ at the contact and in small vicinity of the three phase junction. A thin film formed in this region extends up to a length scale of the order λ (Figure 3). Further, surface roughness also plays a significant role in this inner region. The governing equation of the thin film is the same as eq 5 but is subjected to a Navier slip boundary condition at the wall, whereas the other boundary condition for stress free surface remains the same. The boundary conditions are given by:15 16704

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We scale the axial and lateral lengths as: X = x/bCa1/3 and H = h/bCa2/3 to modify eq 12 in a nondimensional form for the intermediate region as: ⎛ ∂ 3H ⎞ α 2 2−α ⎜ 3 ⎟((H − ε2εsζ ) + εs (H − ε2εsζ ) ) = 3 ⎝ ∂X ⎠

along X axis. It is an implicit indicator of the dynamic contact line and the effects of intermolecular force and again the surface effects are also implicitly related to this parameter. The solution strategy is as follows: when C3 is below a certain threshold value C*3 , it results in a singular solution and the dependent variable H droops down to a value meeting the surface as H → ε1ζ, unable to match the thin film extending to infinity. Whereas another limit is also observed, when C3 exceeds C3*, the solution blows up in the form H → ∞ as X → ∞, which is another extreme where matching to the thin film is impossible. Thus, a proper matching of the solution to the thin (precursor) film is possible at C3*(Kalliadasis and Chang14), which lies between these two extremes. We employ a bisection algorithm to obtain this parameter correct to 10 decimal places (to achieve a better precision). We employ this parameter C3* to obtain the dynamic contact angle using eq 10 for different values of ε and ε1. We compare the inner region solutions for different roughness profiles (randomly generated) corresponding to the same value of ε1 = 0.5 and ε = 10−2 with the corresponding solutions obtained for an idealized smooth surface, as shown in Figure 4. It may be observed that, for a value less than C*3 =

(14)

where εs = 3 λ/bCa is the ratio of the slip length to the characteristic film thickness length scale. In the limit of X → −∞, the intermediate region eq 14 has to match with the outer region solution, whereas in the limit X → X0 it has to match with the inner solution at a point X0. ε2εs represents the scaling of the roughness height as was done in the earlier case where the roughness height (ξrms) scales with bCa2/3. The matching of the inner solution and the intermediate region solution may be ensured by the condition: 1/(2−α)

2/3

31/(2 − α)λη(l → −∞) = bCa 2/3H(X → X 0)

(15)

The limits are interpreted in terms of the matching principle of Van Dyke,63 by expanding H using Taylor’s series expansion and rewriting them in terms of the inner variables to obtain: η(l → −∞) = Dl

(16)

where D = H′(X → 0) valid for O(εs). In the other limit X → −∞, the matching is achieved similar to the precursor film model, which has in this asymptotic limit a dominant quadratic behavior: H→

X2 + D1 as X → −∞ 2

(17)

The modified contact angle due to local slope of the surface roughness may be given by: ⎛ dη dζ ⎞ − ε2 ⎟ = θs Ca1/3⎜ ⎝ dl dl ⎠

(18)

where θs is the static contact angle for smooth surface. The extrapolation of the outer parabolic profile can be done in the same way as in the case of precursor film model resulting in the description of the dynamic contact angle as: θdyn = tan−1( −Ca1/3 −2D1 )

Figure 4. Solution of the contact line for the inner region over a completely wetting substrate for a representative smooth surface (A) and two rough surfaces (B & C). The surfaces B and C are created using the random numbers with ε1 = 0.5, and the nondimensional correlation length (nondimensionalised by bCa2/3) is taken to be of the order of 1. The behavior of the solution for C3 < C*3 meeting the surface (solid lines) and C3 > C3* with a blow-up of the solution (dotted line) is shown for each of these surfaces. The parameter C*3 obtained for smooth surface (A) is −2.08642578125, whereas that for the rough surfaces (B and C) are −0.635986328125 and −3.01513671875, respectively.

(19)

It is important to note that the value of D1 depends on the intermediate region solution, which in turn is influenced by the asymptotic form of the inner region solution.

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RESULTS AND DISCUSSION Complete Wetting. We simulate the inhomogeneities of the surface using random numbers and nondimensionalize those using the RMS value. The topographic variations are considered slow through the specification of Rcorr and restrictions on the random heterogeneities given by eq 2a and 2b. We evaluate the Fourier transforms by using FFT (fast Fourier transform) algorithm. The inner region governing eq 9 for the precursor film model is a third order differential equation, which we further decompose into three coupled first order differential equations and solve using a stiff initial value solver using Runge−Kutta algorithm, with the initial value for H as given by H = (X2/2 + C3) with H′ and H″ as X and 1 respectively. We vary the parameters ε and ε1 to observe the effect of the intermolecular force and the surface roughness respectively, on the contact line dynamics. The parameter C3 is a negative quantity which remains invariant under a translation

−0.635986328125, the profile for H asymptotically meets the surface, whereas beyond this critical limit, rapid growths in the values of H may be observed. For a smooth surface as indicated in the figure, we obtain C3* to be −2.08642578125, whereas for the another roughness profile in consideration we obtain C*3 to be equal to −3.01513671875 keeping all of the other parameters unaltered. Because different (randomly generated) surface roughness profiles lead to different values of C3*, and hence different values of the dynamic contact angle (even by keeping the normalized RMS heights, correlation coefficient, and number of grid points unaltered), it is nearly impossible to predict the trend of the dynamic contact angles unless those are 16705

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considered in a statistical sense and corresponding to large number of simulations for the same sets of parameters. As the surface roughness is mathematically represented by a random function, the net effect on contact angle is represented by statistical mean value of the contact angles (θm) and the span of the deviation of these contact angles (θδ) from the mean value as the respective standard deviations represented by error bars. For each set of parameters, we have performed 1000 simulations to predict these mean and deviation values. The dependence of the mean apparent contact angle θm (in degrees) on the intermolecular force parameter (ε) for the cases of smooth and rough surfaces (ε1 = 0.2) is shown in Figure 5. The results for the smooth surface match to that of

Figure 5. θm (in degrees) as a function ε for smooth and rough surfaces (ε1 = 0.2). For the rough surface, the figure depicts the mean values and the standard deviation based on data obtained for 1000 simulations for completely wetting surface.

the results obtained by Kalliadasis and Chang.14 The effect of roughness as represented by the parameter ε1 leads to decrease in contact angle from a value which corresponds to a smooth surface, as may be observed from Figure 5. It may be also observed from the figure that, for both smooth and rough surfaces (on an average), the contact angle decreases as ε increases. This may be attributed to the increase in intermolecular forces (represented by ε) influencing the wetting of the surface for both smooth and rough surfaces and consequently decreases the contact angle. Further, it may also be observed that any decrease in ε below 10−12 has nearly no influence on the statistics of the contact angle for the rough surfaces, although for the smooth surface it tends to increase the contact angle. Part a of Figure 6 shows the dependence of the dynamic contact angle on the roughness parameter ε1 for different values of ε for a completely wetting substrate. For a given ε, increase in surface roughness results in decrease in contact angle resulting in enhanced wetting. An increase in surface roughness parameter ε1 leads to the increase in the net substrate area (length for a 2D consideration), which leads to an enhancement in the intermolecular interaction. To minimize the interfacial energy, the fluid wets more and consequently decreases the contact angle. Again, the reduction of the channel dimension b leads to increase ε1, which signifies a stronger influence of the surface roughness on the contact line dynamics as the confinement is made progressively narrower. Further, for a creeping motion with very low capillary number (Ca) and high values of ε1, the fluid tends to wet the substrate

Figure 6. (a) θm (in degrees) is the mean of the dynamic contact angle as a function ε1for different values of ε based on the statistical data obtained for 1000 simulations. (b) θδ (in degrees), the deviation from the mean value as a function ε1 for different values of ε shows that the standard deviations are not dependent on the intermolecular force but are rather sensitively dependent on the roughness parameter.

more. It can also be observed from the figure that at higher values of ε1, the contact angle has a larger deviation from the mean value because of the increased deviation of the roughness profile from its mean value. The values of the dynamic contact angle tend to extend up to θ ∼ 50°, which may lead to the possibility of breaking down of the long-wave lubrication approximations.14 Part b of Figure 6 shows the dependence of the standard deviation of the contact angle as a function of roughness parameter ε1. It may be easily verified from the plots that as roughness increases the standard deviation of contact angle also increases, although it is independent of the values of ε. Partial Wetting. The inner region differential equation may be solved by considering different values of D and evaluating the slope near the wall in each case. The value of D is considered such that the inner region solution at the wall corresponds to the slope at the wall as θs for a given slope of the roughness profile (dζ/dl) and the parameter (ε2) as given by eq 18. The solution of intermediate region is evaluated using the boundary condition H′(X → X0) = D and solving the 16706

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We obtain the dynamic contact angle value for εs = 0.1 corresponding to large number of roughness profiles and represent the mean values of the contact angles in part a of Figure 8 for different correlation lengths with the insert

differential equation (eq 30) for different values of C2 using parabolic matching to the outer region. Regarding the contact line dynamics of the partially wetting films, it is important to mention that the same is strongly governed by the product of the parameters ε2 and εs (eq 14). The parameter ε2εs represents a measure of the ratio of the roughness length scale to the channel length scale, normalized by a lateral scale proportional to Ca2/3. An increase in this parameter would imply increments in the roughness in comparison to the channel dimensions or a decrement in the capillary number. However, the matching condition given by eq 15 needs an explicit specification of the parameter εs, which indicates a dependence of the contact angle on the slip length at the three phase contact line. Notably, a substrate may be specified in terms of its static contact angle on a flat surface. Purely for the sake of demonstration without any loss of generality, we consider the same to be θs = 10° as a demonstrative example. We take the slip length parameter εs corresponding to this surface as 0.01 with the specification of α = 1, which is typically the Navier-slip condition. We consider two different roughness profiles for this substrate, with the corresponding parameter ε2εs equal to 0.3. We depict the intermediate regions for these two roughness profiles and the inner region regions in the insert of Figure 6. The matching conditions obtained for the inner region is given by D = −2.442108154296875 and D = −5.950701904296876 for these two profiles respectively (eq 13), whereas the matching condition for the intermediate region is specified by: D1 = −7.499209594726562 and D1 = −21.870809936523436, respectively). The later values dictate the dynamic contact angle, although the inner region satisfies the equilibrium static contact angle of 10°, as may be observed from the inserts of Figure 7.

Figure 8. (a) Dependence of the mean dynamic contact angle θm on the parameter ε2εs for different orders of the nondimensional correlation length Rcorr, with εs = 0.1 (the slip length is kept unchanged) where ε2εs = 0 represents the smooth surface. (b) Variation of the dynamic contact angle on the parameter ε2εs for different orders of slip length parameter, εs, keeping the value of Rcorr = 0.1. For the dominant effect of the intermolecular force, the slip length parameter is taken as εs = 10−4. The dependence of the standard deviation (θδ) of the contact angle as a function of ε2εs is shown in the insert separately to improve the clarity of the figures.

showing the standard deviations. The correlation length, Rcorr, is nondimensionalised by the thin film length scale bCa2/3. Physically, correlation length is an indicator of the length scale over which the generated random roughness profiles in one region are correlated or influenced to those in another region. Two points on the surface separated by a distance larger than the correlation length will have the fluctuations nearly uninfluenced or independent of each other. We vary the order of magnitude of the nondimensional correlation length keeping the value of the slip length to be fixed to investigate its influence on the contact angle and show the corresponding variations in part a of Figure 8. For the correlation length of

Figure 7. Contact line for the intermediate region over a partial wetting substrate for two rough surfaces (A1 & A2) corresponding to εs = 0.01. Numbers of grid points for generating the rough surface are taken to be equal to 120 000 (to resolve both inner and intermediate region solutions), α = 1, and the nondimensional correlation length (nondimensionalised by bCa2/3) is taken to be of the order of 1. The static contact angle for these surfaces is taken to be same as 10°. The insert shows the zoomed view of the inner regions (l and η in abscissa and ordinate) for these two surfaces with the value of the static contact angle. The dynamic contact angles are obtained for the surfaces A1 and A2 are 39.8404 and 54.9380 (in degrees) corresponding to D1 values of −7.4992 and −21.8708, respectively. 16707

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condition. However, the decrease in contact angle for εs = 10−4 (which corresponds to the dominant effects of intermolecular force) is almost equal to that obtained for the precursor film model, which shows the equivalence of these models in the limiting case.

order less than 1, the roughness influences the inner region. As the fluid is more exposed to the solid surface, the intermolecular force dominates and reduces the contact angle. As the order of the correlation length increases, its effect on the contact angle decreases. For the correlation length of O(1), (O() refers to order of magnitude) the effects of the intermolecular force become the least significant as the contact angle remains nearly constant. For Rcorr ∼ O(10), the contact angle remains virtually constant but with increased value in comparison to O(1). We further try to observe the variation of the dynamic contact angle by varying the product ε2εs in part b of Figure 8. It may be observed that the increase in slip parameter results in decrease in the contact angle. From a mathematical viewpoint, it may be inferred that the intermediate region solution satisfies the form H → εsη(l → −∞) from eq 15 in the limit of X → 0. As the solution is marched toward X → −∞, the governing equation of the intermediate equation (given by eq 10) behaves as ∂3H/∂X3 → 0, which implicates a quadratic dependence of the form of H → X2/2 + D1 (eq 17). With higher value of the slip length, the parameter εs increases linearly and the initial value of the variable H(X → X0) also increases. The monotonic behavior of the eq 14 (as X decreases, H increases) ensures that the magnitude of the quadratic matching parameter D1 decreases with increase in initial value of H and consequently results in decrease in contact angle with increase in slip length or the parameter εs. Interestingly, one of the recent studies proposes the equivalence of the slip model with the precursor film model.63 It has been shown that, the quasi-static limit of a slip model is equivalent to that of a precursor film model when εl ̃ = εs where ε is the parameter, which we have considered for the precursor film model, εs is the nondimensional slip length and the parameter l ̃ is defined as:64 ln l ̃ =

∫1

(α − 1)2 ⎞⎟ ⎜⎜ − ⎟d α α 3 1 + K (α ) ⎠ ⎝α

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CONCLUSIONS We have investigated the effects of surface topography on the contact line dynamics in a narrow confinement with different wetting characteristics of the substrate, namely, the completely wetting and partially wetting cases. We have modeled the surface roughness as random function. The inner solution resolves the forces close to the wall by including surface tension, intermolecular, and viscous effect, in a topographically varying environment, although in a quasi-steady sense. The outer region represents the shape of the bulk meniscus. We have asymptotically matched these two regions, either directly or with the help of an intermediate region, to yield a complete description of the contact line. We have obtained dynamic contact angle by appropriately extrapolating the bulk meniscus toward the wall. We have represented the variations in the contact angle using the pertinent statistical measures (mean and standard deviation) for a large number of random surface profiles. The present model should be judiciously used considering the validity of the approximations. The dominant curvature of the meniscus is determined by the stress balance equation near the centerline as given by eq 3 which governs the outer solution of the meniscus. The scale involved with the outer region is of the order of the channel width. The first order correction is of the order of Ca1/3 where roughness starts playing role. In strict terms, the symmetry at the channel centerline as given by condition (4b) would not be valid. However, for small surface roughness, it may be approximated to be symmetric because of the disparate scales involved with the bulk meniscus and that of its effects on the modifications of contact angle. Slow variation in the topography is considered so that the contact line is not trapped in sharp corners of the roughness element. The average velocity (U) of the meniscus has also been considered constant which is valid only for shallow topographies. In essence, although U may not be constant, the slowly varying contact angle is captured for the creeping motion (with very low capillary number Ca) of the meniscus, through the functional dependence on Ca. The motion of the contact line has been, hence, represented in only an average sense, without considering the explicit temporal variation. The results are correct to the order of O(Δt2) and hence the contact angle should be interpreted for very small time intervals. As the present study does not consider detailed dynamics of the contact line (because of the approximations of constant average velocity U), further studies should be performed addressing full dynamics of the problem to understand the consequence of roughness on wetting condition (enhancement or inhibition) through full scale numerical computations in three dimensions. It is important to mention in the present context that a liquid−gas interface moving over the rough surface elements may not be successful in displacing the gas over the entire corrugated region. This may lead to evacuated regions (gas filled) beneath the moving liquid film. These roughness elements trapping the gas phase may be considered locally as small grooves with V-shaped structures. The angle of inclination of these groove walls with the horizontal is φ. The necessary condition stating that these grooves will be wet

∞⎛1

(33)

where the parametric function K(α) depends on the choice of the form of the disjoining pressure. In the present case, we have chosen the parameter l ̃ = 43 and considering ε = 10−5 for precursor film model, the nondimensional slip length εs may be obtained as 10−4. From another viewpoint, if the van der Waals forces are dominant near the contact line, the slip length λ can be modified by:65 λ = (|A|/6πσlg)1/2 and consequently the parameter εs becomes 3/Ca2/3b(|A|/6πσlg)1/2, which also yields a value 10−4. Corresponding to case εs = 10−4, we obtain the variation of the dynamic contact angle with ε2εs in part b of Figure 8. The contact angle decreases from a reference value corresponding to the dynamic contact angle on the flat surface (ε2εs = 0) for all of the cases. It may be observed that the contact angle decreases with increasing roughness and also decreases with increase in slip length. However, the decrease in contact angle is not substantial in comparison to the case of the fully wetting surface as was studied in the previous case except for the case of εs = 10−4. The thin film region only extends to few nanometers for the partial wetting surface, whereas for the complete wetting substrate, it does extend far to infinity in the form of precursor film. Mathematically, with the presence of a long precursor film, the intermolecular force directly comes into the thin film equation in the form of the disjoining pressure, whereas for partially wetting substrate, the effect of the intermolecular forces is only linked by the slip boundary 16708

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is given by θs < (π − 2ϕ) where θs is the contact angle of interface is in static condition. In the purview of the present study, we consider the static contact angles to be small enough (satisfying the above criterion), so as to enable wetting or nearly wetting behavior resulting in all of the roughness elements to be considered wet for the theoretical analysis. Further, we assume slow variation of the topographical features for the analysis to be valid. Nevertheless, the abstraction of the dynamic contact angle, as obtained from the present study, may turn out to be significant in simulating the capillary filling dynamics using a reduced order model66,67 accounting for the surface roughness, interfacial slip, and different wetting behavior of the substrate in an integrated framework. Additional effects of interfacial electrokinetics on the contact line dynamics68 over rough surfaces may also be investigated by extending the present work. Combined effect of surface roughness and wettability can be further extended through the prescription of this dynamic contact angle as a function of the relevant roughness and slip parameters, without explicitly resolving the topographical details of the substrate. Random roughness function plays the role of a synthetic microscope to visualize topographical features of the interface morphology and represent the artifacts of real surface produced by different microfabrication processes. Further, roughness induced phase transition results in vapor dispersed layers in the form of the nanobubbles at the surface, which may play crucial role in altering the dynamics of the three phase contact line. Such effects could also be modeled using the present approach by modeling the nanobubble dispersed layer with an effective slip. Future studies on the hysteresis of the contact lines for an advancing and receding contact lines in confined geometries are currently under progress.

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ASSOCIATED CONTENT

S Supporting Information *

Mathematical derivation of the liquid pressure and asymptotic matching of the inner region of the complete wetting case with the solution of outer region. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], tel: +91(3222)282990. Notes

The authors declare no competing financial interest.

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snapshot of the random rough profile is considered in a quasi-steady sense. This would incur an error of the order of O(Δt2) but capture the essence of the contact line to evaluate the dynamic contact angle as a function of the capillary number Ca (which in turn is a function of the average velocity U). As, the walls are not flat, the transformation also alters the roughness profiles obtained at different instants. The roughness term in the reference frame moving with meniscus is considered to be a function of both axial coordinate X and time t, that is ζ = ζ(bCa1/3X + x0 + Ut). (60) Bretherton, F. P. The motion of long bubbles in tubes. J. Fluid Mech. 1961, 10, 166−188. (61) Dressaire, E.; Courbin, L.; Crest, J.; Stone, H. A. Inertia dominated thin-film flows over microdecorated surfaces. Phys. Fluid 2010, 22, 073602. (62) Kalliadasis, S.; Bielarz, C.; Homsy, G. M. Steady free-surface thin film flows over topography. Phys. Fluids 2000, 8, 1889−1898. (63) Van Dyke, M. Perturbation Methods in Fluid Mechanics; Academic, 1964. (64) Savva, N.; Kalliadasis, S. Dynamics of moving contact lines: A comparison between slip and precursor film models. Euro Phys. Lett. 2011, 94, 64004. (65) Eggers, J. Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 2004, 93, 094502−4 (2004). Eggers, J. Existence of receding and advancing contact lines Phys. Fluids 2005, 17, 082106. de Gennes, P. G.; Hua, X.; Levinson, P. Dynamics of wetting: Local contact angles. J. Fluid Mech. 1990, 212, 55. (66) Chakraborty, S. Dynamics of capillary flow of blood into a microfluidic channel. Lab Chip 2005, 5, 421−430. (67) Chakraborty, S. Electroosmotically driven capillary transport of typical non-Newtonian biofluids in rectangular microchannels. Anal. Chim. Acta 2007, 605, 175−184. (68) Chakraborty, D.; Chakraborty, S. Interfacial phenomena and dynamic contact angle modulation in microcapillary flows subjected to electroosmotic actuation. Langmuir 2008, 24, 175−9449.

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