Droplet Morphologies on Particles with Macroscopic Surface

Jan 7, 2006 - Droplet size and contact angle were found to generally have a stronger effect on surface coverage than particle surface roughness. Becau...
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Langmuir 2006, 22, 917-923

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Droplet Morphologies on Particles with Macroscopic Surface Roughness Frantisˇek Sˇ teˇpa´nek*,† and Pavol Rajniak‡ Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom, and Merck & Company, Inc., Research Laboratories, West Point, PA 19486-0004. ReceiVed July 13, 2005. In Final Form: NoVember 1, 2005 The equilibrium configuration of liquid droplets on the surface of macroscopically rough solid particles was determined by numerical simulations using the volume-of-fluid (VOF) method. The fractional surface coverage of the particle as a function of the droplet size, equilibrium contact angle, and the particle surface roughness amplitude and correlation length has been systematically investigated. Droplet size and contact angle were found to generally have a stronger effect on surface coverage than particle surface roughness. Because of droplet coalescence, a relatively large variation in surface coverage was observed for any given total liquid volume, particularly for larger values of the equilibrium contact angle.

1. Introduction The wetting of macroscopically rough irregular particles is encountered in many industrial processes such as coating1, wet granulation,2,3 or the manufacture of particle-reinforced composite materials. In wet granulation, for example, the degree of particle coverage by a binder solution determines the rate of agglomeration. In most of the above-mentioned processes, particles are exposed to a spray of fine liquid droplets. Depending on the relative droplet size and the particle surface morphology and wettability, partial or total coating can be obtained. In this paper, we address the practically important as well as theoretically interesting question of particle coatability; that is, we seek to find the functional relationship between the degree of surface coverage and the volume of liquid present on the particle, knowing the particle morphology and the equilibrium contact angle. In the absence of gravity or other volume forces, a droplet deposited on a flat solid surface will assume the equilibrium shape of a spherical cap. In that case, the relationship between the droplet volume, V, the wetted area, A, and the equilibrium contact angle, θeq, can be expressed analytically.4 If the underlying surface is microscopically rough (In the context of this work, the term macroscopic roughness refers to surface morphological features occurring at length scales comparable to the droplet diameter, while microscopic roughness refers to surface heterogeneities at length scales much smaller than the droplet size), the apparent equilibrium contact angle will be modified according to Wenzel’s law,5,6 but the droplet shape will still be that of a spherical cap. A general classification of liquid morphologies on * Corresponding author. Tel.: +44 20 7594 5608; fax: +44 20 7594 5604; e-mail: [email protected]. † Imperial College London. ‡ Merck & Co., Inc.

(1) Teunou, E.; Poncelet, D. Batch and Continuous Fluid Bed Coating Review and State of the Art. J. Food Eng. 2002, 53, 325-340. (2) Link, K. C.; Schlu¨nder, E. U. Fluidized Bed Spray Granulation Investigation of the Coating Process on a Single Sphere. Chem. Eng. Process. 1997, 36, 443-457. (3) Zhang, D.; Flory, J. H.; Panmai, S.; Batra, U.; Kaufman, M. J. Wettability of Pharmaceutical Solids: Its Measurement and Influence on Wet Granulation. Colloids Surf., A 2002, 206, 547-554. (4) Clarke, A.; Blake, T. D.; Carruthers, K.; Woodward, A. Spreading and Imbibition of Liquid Droplets on Porous Surfaces. Langmuir 2002, 18, 29802984. (5) Bico, J.; Thiele, U.; Que´re´, D. Wetting of Textured Surfaces. Colloids Surf. A 2002, 206, 41-46.

planar substrates has been proposed.7 If the substrate is not planar, however, determining the equilibrium configuration of a liquid generally requires a numerical solution; the problem of finding an equilibrium configuration of a droplet on a fiber8 can serve as a typical example. In principle, any method capable of free interface tracking can be used for finding the equilibrium configuration of a droplet on a nonplanar surface. Examples include a finite element-based method9 or discrete particle-based methods such as latticeBoltzmann10 or dissipative particle dynamics.11 Here the volumeof-fluid (VOF) method12,13 is used. We previously applied the VOF method for the simulation of various situations involving free interface propagation in spatially complex morphologies, such as capillary condensation,14 bubble nucleation and coarsening in porous media,15 or droplet spreading and solidification in particle agglomerates.16 In the present work, the problem of finding the equilibrium liquid configuration on the surface of irregular particles is addressed. Complex droplet morphologies and wetting transitions have been observed even on geometrically relatively simple domains such as planar substrates chemically patterned by stripes18 (6) De Coninck, J.; Ruiz, J.; Miracle-Sole´, S. Generalized Young’s Equation for Rough and Heterogeneous Substrates: A Microscopic Proof. Phys. ReV. E 2002, 65, 036139. (7) Neimark, A. V.; Kornev, K. G. Classification of Equilibrium Configurations of Wetting Films on Planar Substrates. Langmuir 2000, 16, 5526-5529. (8) McHale, G.; Newton, M. I. Global Geometry and the Equilibrium Shapes of Liquid Drops on Fibers. Colloids Surf., A 2002, 206, 79-86. (9) Brakke, K. A. The Surface Evolver. Exp. Math. 1992, 1, 141-165. (10) Raiskinma¨ki, P.; Koponen, A.; Merikoski, J.; Timonene, J. Spreading Dynamics of Three-Dimensional Droplets by the Lattice-Boltzmann Method. Comput. Mater. Sci. 2000, 18, 7-12. (11) Clark, A. T.; Lal, M.; Ruddock, J. N.; Warren, P. B. Mesoscopic Simulation of Drops in Gravitational and Shear Fields. Langmuir 2000, 16, 6342-6350. (12) Rider, W. J.; Kothe, D. B. Reconstructing Volume Tracking. J. Comput. Phys. 1998, 141, 112-152. (13) Bussmann, M.; Mostaghimi, J.; Chandra, S. On a Three-Dimensional Volume Tracking Model of Droplet Impact. Phys. Fluids 1999, 11, 1406-1417. (14) Sˇ teˇpa´nek, F.; Marek, M.; Adler, P. M. Modeling Capillary Condensation Hysteresis Cycles in Reconstructed Porous Media. AIChE J. 1999, 45, 19011912. (15) Sˇteˇpa´nek, F.; Marek, M.; Adler, P. M. The Effect of Pore Space Morphology on the Performance of Anaerobic Granular Sludge Particles Containing Entrapped Gas. Chem. Eng. Sci. 2001, 56, 467-474. (16) Sˇteˇpa´nek, F.; Ansari, M. A. Computer Simulation of Granule Microstructure Formation. Chem. Eng. Sci. 2005, 60, 4019-4029. (17) Schwartz, L. W. Hysteretic Effects in Droplet Motions on Heterogeneous Substrates: Direct Numerical Simulation. Langmuir 1998, 14, 3440-3453.

10.1021/la051901u CCC: $33.50 © 2006 American Chemical Society Published on Web 01/07/2006

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918 Langmuir, Vol. 22, No. 3, 2006

Figure 1. Particle morphologies obtained by varying the dimensionless surface roughness amplitude, a, and correlation length, L.

or rectangular patches,17 or on linear wedge-shaped grooves19 and their networks.20 A variety of equilibrium droplet morphologies on particles with macroscopic surface roughness can therefore be expected.

2. Methodology 2.1. Particle Shape. The particle shape is described by a Gaussian-correlated closed random surface characterized by a mean radius of gyration, rg, an amplitude, a, and a correlation length, L, and generated in the following way. Let X(u), u ∈ [0; 2π] × [-π/2; +π/2] be a field of independent normally distributed random variables. A field of Gaussian-correlated random variables is obtained from X by applying a linear filter

Y(u) )

∑

exp(V2/L2)X(u + v)

(1)

V∈[0;L]2

and renormalization, where V ) |v| is a distance calculated using the geodesic metric. The particle surface is then obtained by modulating a sphere with radius rg by the Gaussian-correlated random surface Y(u). In subsequent notation, a and L will denote dimensionless quantities scaled by rg. Examples of particle shapes obtained by systematically varying the surface roughness amplitude and correlation length are shown in Figure 1. Each combination (a, L) defines a family of particles where the exact shape of each individual particle depends on the random initialization of the uncorrelated field X(u). Unless stated otherwise, simulation results reported in the following sections represent averaged quantities obtained from a random realization of several particles for each (a, L) pair. The mean radius of gyration, rg, amplitude, a, and correlation length, L, are convenient (18) Brinkmann, M.; Lipowsky, R. Wetting Morphologies on Substrates with Striped Surface Domains. J. Appl. Phys. 2002, 92, 4296-4306. (19) Brinkmann, M.; Blossey, R. Blobs, Channels and “Cigars”: Morphologies of Liquids at a Step. Eur. Phys. J. E 2004, 14, 79-89. (20) Dussaud, A. D.; Adler, P. M.; Lips, A. Liquid Transport in the Networked Microchannels of the Skin Surface. Langmuir 2003, 19, 7341-7345.

particle shape descriptors, as they can be readily evaluated from two-dimensional21 or three-dimensional22 digital images of real particles. 2.2. Equilibrium Droplet Configuration. At equilibrium, a liquid droplet in contact with a solid surface must satisfy two conditions: (i) the equilibrium contact angle must be satisfied at all three-phase contact lines, and (ii) the mean curvature of the gas-liquid interface must be constant. (A difference in interface curvature would cause a pressure gradient to exist within the liquid droplet, and this would induce flow.) The problem of finding the equilibrium droplet shape can be approached either as a fluid dynamics problem or as a shape transformation problem. In the former approach, the equilibrium droplet configuration is obtained as the asymptotic solution of droplet spreading dynamics for long times, that is, by solving the Navier-Stokes equations with appropriate boundary conditions17,23 or by a discrete particlebased simulation of spreading.24 In the latter approach, the droplet interface motion is governed by constitutive equations that satisfy the above-mentioned equilibrium conditions upon convergence, without resolving the velocity field within the droplet. The fluid dynamics approach is necessary in situations where there is an important contribution of inertia (e.g., droplet impact at high Weber numbers) in the presence of dominant volume forces (e.g., gravity), or where internal transport (heat transfer, convection-diffusion) within the droplet is of interest. In situations where the characteristic time of the liquid-gas interface relaxation to a shape of constant mean curvature is sufficiently smaller than the characteristic time of three-phase contact line motion, such as in the spreading of small, viscous droplets24 or in late stages of spreading,25 the fluid dynamics problem can be reduced to the shape transformation problem. Spreading is then governed by a constitutive equation for contact line velocity as a function of the instantaneous contact angle, and two-phase interfaces are relaxed to a shape of constant mean curvature while conserving the droplet volume. On a planar substrate, the two-phase interface position can be found analytically.4 On a complex substrate, a numerical procedure is required. The shape transformation approach is taken in this work, and the VOF method is used for the numerical tracking of interface positions. The morphology of the solid particle and the spatial distribution of the liquid phase are encoded using the solid and liquid VOF functions fS and fL, respectively, on a grid of N3 cubic volume elements with spatial discretization step h. The VOF function can have values fj ∈ [0;1], j ) S, L, and interface points satisfy 0 < fj < 1, j ) S, L. The interface curvature, κL, at the gas-liquid interface points is calculated from its definition

κL ) -∇‚nL

(2)

where nL is the interface normal vector oriented from the liquid to the gas phase. The normal vector is calculated from the VOF function at each liquid-gas interface point according to the formula (21) Bowman, E. T.; Soga, K.; Drummond, T. Particle Shape Characterization Using Fourier Analysis. Geotechnique 2001, 51, 545-554. (22) Garboczi, E. J. Three-Dimensional Mathematical Analysis of Particle Shape Using X-ray Tomography and Spherical Harmonics: Application to Aggregates Used In Concrete. Cem. Concr. Res. 2002, 32, 1621-1638. (23) Alleborn, N.; Raszillier, H. Spreading and Sorption of a Droplet on a Porous Substrate. Chem. Eng. Sci. 2004, 59, 2071-2088. (24) de Ruijter, M. J.; Blake, T. D.; De Coninck, J. Dynamic Wetting Studied by Molecular Modeling Simulations of Droplet Spreading. Langmuir 1999, 15, 7836-7847. (25) Warren, P. B. Late Stage Kinetics for Various Wicking and Spreading Problems. Phys. ReV. E 2004, 69, 041601.

Liquid Droplets on Macroscopically Rough Solids

nL ) -

∇f˜L |∇f˜L|

Langmuir, Vol. 22, No. 3, 2006 919

(3)

where ˜fL is a smoothed VOF function obtained from fL by the application of a linear smoothing filter.12 The contact angle at three-phase points (i.e., points where fL > 0 and simultaneously fS > 0) is calculated from the interface normal vectors

cos θ ) nS‚nL

(4)

where nS is obtained from the solid VOF function by applying eq 3 to fS rather than fL. In regular two-phase points, the gradient ∇f˜L in eq 3 is evaluated by a symmetric second-order finite i-1 difference formula, that is, ∂f˜L/∂x ≈ (f˜ i+1 L -f˜ L )/2h (etc. for y and z); in the three-phase points the forward or backward finite difference formula has to be used because ˜fL is undefined in the solid phase (this is equivalent to extrapolating the interface position into the solid phase as indicated in Figure 2). Starting from an arbitrary initial droplet shape, the equilibrium configuration of a droplet on the particle surface is found by an iterative procedure where, in every step, the value of the liquid VOF function fL at every interface point is updated by δfL according to the following rules: 1. At gas-liquid interface points, δfL ) R(κL - κjL) where κL is calculated according to eq 2, and κjL is the average interface curvature of the droplet; 2. At three-phase points, δfL ) β(cos θeq - cos θ) where θeq is the equilibrium contact angle, and θ is calculated from eq 4; 3. The total volume of the liquid must be conserved, that is, ∑δfL ) 0. Above, R and β are numerical parameters that influence the stability and speed of the convergence. The iteration stops when max|δfL| e , where  is a required precision. An illustration of the convergence of two droplets to an equilibrium configuration according to the iterative procedure described above is shown in Figure 3. The initial conditions are spherical droplets of dimensionless diameter d ) 0.53 (the scaling factor is 2rg where rg is the mean radius of gyration of the particle). Note that, before the two spreading droplets touched each other and merged, each was tending toward a different mean interface curvature due to different local solid-phase morphology. It should also be stressed that, although the intermediate shapes might imply otherwise, this procedure does not represent a dynamic simulation of spreading but only a shape transformation tending toward a configuration that satisfies the equilibrium contact angle and constant mean curvature. 2.3. Surface Coverage Function. Let us define the particle surface coverage, ψ, as the fraction of the total particle surface occupied by the liquid phase, that is

Awet ψ≡ Atot

(5)

(where Awet is the contact area between the liquid and the solid phase, and Atot is the total particle surface area) and the liquidto-solid ratio, xLS, as the ratio of the total droplet volume to the particle volume, that is

xLS ≡

VL NdVd ) VS VS

Figure 2. Detail of the local condition at a three-phase contact line and the calculation of contact angle from interface normal vectors.

(6)

where Nd is the number of deposited droplets, Vd is the single droplet volume, and VS is the particle volume. The functional dependence of ψ on xLS was determined computationally for each given combination of particle shape (a, L), equilibrium contact angle θeq, and relative droplet diameter d, by depositing

Figure 3. Sequence of intermediate droplet shapes during iteration toward the equilibrium configuration according to the algorithm described in section 2.2. Particle properties: a ) 0.5, L ) 0.27; initial droplet diameter: d ) 0.53; equilibrium contact angle: cos θeq ) 0.95. The iteration count is denoted by t′.

the required number of droplets in random positions on the particle surface and then finding their equilibrium morphology as described in section 2.2 above. At least 10 different realizations of the random initial droplet position were used for every xLS value in order to sample the particle surface and ensure good statistics. The number of droplets was systematically increased in order to realize xLS ratios in the range of [0;1] in steps not larger than 0.1. An empirical trend function of the form

ψ ) 1 - exp(-kxLS)

(7)

was fitted to the complete data set of (xLS,ψ) pairs (typically about 80-100 points) by nonlinear least-squares regression. The dependence of the parameter k, which then fully specifies the surface coverage function ψ(xLS), on the particle shape (a, L), contact angle θeq, and droplet size d is discussed in detail in the sections below.

3. Results and Discussion 3.1. Model Verification. Let us first investigate the accuracy of the numerical scheme and the dependency of the results on spatial discretization. This will be done by comparing the values of a wetted area obtained from a VOF simulation at several spatial resolutions with those of areas calculated analytically for the case of a droplet on a planar substrate where an analytical solution is known.4 Neglecting deformation by gravity, a droplet on a planar surface will assume the asymptotic shape of a spherical cap. The wetted area, A, can therefore be calculated from the

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920 Langmuir, Vol. 22, No. 3, 2006

Figure 5. Dependence of the particle surface coverage on the total liquid/solid ratio for a particle with L ) 0.40 and a ) 0.2 and the following combinations of droplet size and contact angle: (a) d ) 0.33, cos θeq ) 0.95; (b) d ) 0.53, cos θeq ) 0.95; (c) d ) 0.33, cos θeq ) 0.50; (d) d ) 0.53, cos θeq ) 0.50. The data points are simulation results, and the solid and dashed lines are trendlines for cos θeq ) 0.95 and 0.50, respectively.

Figure 4. (a) Comparison of analytically (from eq 8) and numerically obtained contact area of a droplet on a planar substrate as a function of droplet volume for two values of contact angle, cos θeq ) 0.25 and 0.80. (b) Convergence toward the numerical solutions shown in panel a.

droplet volume, V, and the contact angle, θeq, by

A)

[

3Vπ0.5 sin3 θeq

2 - 3 cos θeq + cos3 θeq

]

2/3

(8)

The analytical and numerical solutions are compared in Figure 4 for two values of equilibrium contact angles and four spatial resolutions (droplet diameter before spreading of 6, 8, 12, and 16 units, respectively). In this parameter range, the analytical and numerical solutions are in a good agreement. Simulations reported in the following sections were obtained on a spatial grid of N3 ) 803; the smallest droplet size used in the simulations had a diameter of 7 mesh points, with the largest being 22 mesh points. 3.2. Effect of Particle Shape on Surface Coverage. Let us first look at the liquid morphologies and surface coverages on two different particle shapes, representing the limiting cases of a smooth, nearly spherical particle with a ) 0.2 and L ) 0.40, and a rough particle with a ) 0.5 and L ) 0.13 (cf. Figure 1). The dependence of surface coverage on the liquid-to-solid ratio ψ(xLS) for the smooth particle is plotted in Figure 5 for four combinations of contact angle (hydrophilic case with cos θeq ) 0.95 and a hydrophobic one with cos θeq ) 0.50) and droplet size (d ) 0.33 and d ) 0.53). The simulation data points as well as the fitting function (eq 7) are shown in Figure 5 to illustrate the relatively wide spread of surface coverage ψ for a given xLS value; however, the resulting standard deviation around the trendline based on all ∼80 data points was no more than 3-5%. Several trends can be observed in Figure 5. First of all, the function ψ(xLS) is initially nearly linear, that is, the surface coverage is simply proportional to the number of droplets deposited on the surface; however, as the liquid volume ratio, and thus the number of droplets, increases, the droplets begin to coalesce on the particle surface as their separation distance

Figure 6. Equilibrium liquid morphologies on a smooth particle (L ) 0.40, a ) 0.2) for four combinations of droplet size and contact angle, corresponding to cases a-d from Figure 5. The initial conditions were realized by the random deposition of five droplets (the deposition locations are the same in each of the four cases).

becomes shorter. The consequence is a slowing down of the rate at which ψ increases with xLS. This effect is most prominent in case d in Figure 5, that is, larger droplets and larger equilibrium contact angle. The effect of contact angle is interesting in that, although a larger equilibrium contact angle θeq initially allows the droplets to be closer to each other without coalescence, once they do coalesce, the new liquid-solid contact area is much smaller than the combined contact area of the original droplets in the case of a hydrophobic particle. The effect of droplet coalescence on the surface coverage of a smooth particle is illustrated in detail in Figure 6 where, in each row, the hydrophobic (left) and hydrophilic (right) cases are directly compared for the same particle and the same liquid volume. The top row shows the case of larger droplets (d ) 0.53), and the bottom row shows smaller ones (d ) 0.33). In both cases, it can be clearly seen that, as the particle becomes more hydrophilic, the initially independent liquid clusters spread further and coalesce. Similar analysis has been performed in the case of a rough particle (a ) 0.5, L ) 0.13); the resulting dependence on ψ(xLS) is plotted in Figure 7, again for four combinations of contact angle and droplet size. Qualitatively, the same trends seen in the

Liquid Droplets on Macroscopically Rough Solids

Figure 7. Dependence of the particle surface coverage on the total liquid/solid ratio for a particle with L ) 0.13 and a ) 0.5 and the following combinations of droplet size and contact angle: (a) d ) 0.33, cos θeq ) 0.95; (b) d ) 0.53, cos θeq ) 0.95; (c) d ) 0.33, cos θeq ) 0.50; (d) d ) 0.53, cos θeq ) 0.50. The data points are simulation results, and the solid and dashed lines are trendlines for cos θeq ) 0.95 and 0.50, respectively.

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Figure 9. Dependence of the exponent k on the surface roughness correlation length L for d ) 0.40, cos θeq ) 0.95, and two values of amplitude a, as indicated in the graph. Table 1. Values of the Exponent k as a Function of Surface Roughness Correlation Length, L, the Relative Droplet Diameter, d, and the Contact Angle, θeq, for Surface Roughness Amplitude a ) 0.2 a

L

d

cos θeq

k

0.2 0.2

0.13 0.40

0.53 0.53

0.95 0.95

1.382 ( 0.037 1.254 ( 0.030

0.2 0.2

0.13 0.40

0.53 0.53

0.50 0.50

0.750 ( 0.016 0.686 ( 0.014

0.2 0.2

0.13 0.40

0.33 0.33

0.95 0.95

2.183 ( 0.049 1.806 ( 0.046

0.2 0.2

0.13 0.40

0.33 0.33

0.50 0.50

1.008 ( 0.028 0.881 ( 0.028

Table 2. Values of the Exponent k as a Function of Surface Roughness Correlation Length, L, the Relative Droplet Diameter, d, and the Contact Angle, θeq, for Surface Roughness Amplitude a ) 0.5

Figure 8. Equilibrium liquid morphologies on a rough particle (L ) 0.13, a ) 0.5) for four combinations of droplet size and contact angle, corresponding to cases a-d from Figure 7. The initial conditions were realized by the random deposition of five droplets.

case of the smooth particle are observed. The order in which the four cases are arranged is the same, but a careful comparison of Figure 5 and Figure 7 reveals some important differences: namely, that the uppermost curve (cos θeq ) 0.95, d ) 0.33) gives a systematically higher coverage in the case of the rough particle, and that the change in surface coverage caused by changing the droplet size under otherwise identical conditions (i.e., the apparent “distance” from a to b or from c to d in the graphs) is also larger in the case of the rough particle. Let us look in detail at the droplet morphologies shown in Figure 8 in order to try to understand these trends. It seems that the “retraction” of the droplet after coalescence occurs to a much smaller extent on a rough particle, probably because the richness of the supporting solid morphology provides enough opportunities for the equilibrium contact angle to be satisfied locally without the need for the contact line to retract too far; in fact, the contour of the droplet is nowhere near circular on the rough particle, even for the hydrophobic case. On a rough particle, the wetted area after coalescence therefore remains closer to the sum of the wetted areas before coalescence. The values of the exponent k appearing in the coverage function (eq 7) corresponding to the eight situations discussed above are

a

L

d

cosθeq

k

0.5 0.5

0.13 0.40

0.53 0.53

0.95 0.95

1.461 ( 0.036 1.343 ( 0.041

0.5 0.5

0.13 0.40

0.53 0.53

0.50 0.50

0.747 ( 0.014 0.727 ( 0.018

0.5 0.5

0.13 0.40

0.33 0.33

0.95 0.95

2.302 ( 0.039 1.836 ( 0.027

0.5 0.5

0.13 0.40

0.33 0.33

0.50 0.50

1.201 ( 0.021 1.017 ( 0.020

given in Tables 1 and 2 for the smooth and rough particle, respectively. In addition to that, Tables 1 and 2 also contain the values of k for the “mixed” cases of (a ) 0.2; L ) 0.13) and (a ) 0.5; L ) 0.40), which were not plotted in Figures 5 and 7. These sixteen cases represent the “corners” of a hypercube in a four-dimensional (a, L, d, cos θeq) parameter space. While the tables give a good indication of the parametric sensitivity of the exponent k on each of the four parameters, it is also interesting to investigate the shape of the function k(a, L, d, cos θeq) with respect to the individual parameters within the hypercube, so that appropriate interpolation can be done when surface coverage for a specific combination of parameters needs to be estimated. Let us start by investigating the projection of the k(a, L, d, cos θeq) function into the (a, L) plane by fixing an intermediate droplet size value of d ) 0.40, keeping cos θeq ) 0.95, and systematically changing the values of the amplitude, a, and correlation length, L, of the particle surface roughness following the “frame” of Figure 1, that is, 0.13 e L e 0.40 for a ) 0.1 and a ) 0.5 (results are shown in Figure 9), and 0.1 e a e 0.5 for L ) 0.13 and L ) 0.40 (Figure 10). The graph in Figure 9 reveals that linear interpolation with respect to L will be sufficient,

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922 Langmuir, Vol. 22, No. 3, 2006

Figure 10. Dependence of the exponent k on the surface roughness amplitude a for d ) 0.40, cos θeq ) 0.95, and two values of the correlation length L, as indicated in the graph.

Figure 11. Dependence of the exponent k on the relative droplet diameter d for a particle with a ) 0.5, L ) 0.27, and two values of the equilibrium contact angle, as indicated in the graph.

at least in this region of parameters. The exponent k is a decreasing function of the correlation length, and, for a larger amplitude (a ) 0.5), the decrease is stronger. The decrease of k with L means that roughness occurring over shorter length-scales favors surface coverage, the reason probably being the lesser extent of contact line retraction already discussed above. Turning to Figure 10, where k is plotted as a function of the amplitude, the trends become more interesting: a maximum with respect to a is apparent for both values of L, and thus for all intermediate values of L as well, given the linear dependence on L just shown in the previous figure. This means that increasing the surface roughness amplitude initially improves surface coverage, but once the amplitude becomes too large, surface coverage starts to decrease again, most probably due to “peaks” of solid phase, which remain unwetted even at relatively large volumes of the liquid. In the following sections, the effect of the remaining two parameterss the droplet size and the contact angleswill be investigated in detail. 3.3. Effect of Droplet Size on Surface Coverage. A parametric study with respect to d was carried out for a particle with intermediate roughness (a ) 0.5 and L ) 0.27) and two values of contact angle: a hydrophilic particle with cos θeq ) 0.95 and a hydrophobic one with cos θeq ) 0.50. The results of the simulations are summarized in Figure 11. The parameter k is a monotonically decreasing function of the droplet size for both contact angles; however, its dependence on contact angle is much stronger in the hydrophilic case (cos θeq ) 0.95). The graph also nicely illustrates that, in the asymptotic case of d f 0 and hydrophilic particle, the exponent k tends to infinity (meaning that practically full coverage can be achieved even at low liquid volume ratios, xLS), whereas, for the hydrophobic particle, the asymptote seems to be a finite value of k. This is consistent with

Figure 12. Dependence of the exponent k on equilibrium contact angle for a particle with a ) 0.5, L ) 0.27, and two values of the relative droplet size, d ) 0.33 and 0.53, as indicated in the graph.

the empirical knowledge that finer sprays generally produce better particle coating, but that uniform coating of hydrophobic particles is difficult to obtain, regardless of droplet size, because of droplet coalescence and dewetting. 3.4. Effect of Contact Angle on Surface Coverage. Finally, a parametric study with respect to the equilibrium contact angle was carried out. The contact angle was systematically varied in the range of cos θeq ) 0.25-0.99 for two values of relative droplet diameter d ) 0.33 and 0.53, and fixed particle surface roughness a ) 0.5 and L ) 0.27, as in the previous section. The exponent k as a function of contact angle is plotted in Figure 12. (The curves in Figures 11 and 12 can be thought of as orthogonal projections of the parametric surface k(d, θeq) at the specified values of the two parameters.) The exponent k is a monotonically increasing function of cos θeq, and it is interesting to observe that, for more hydrophobic particles (cos θeq < ∼0.6), the effect of droplet size is relatively small, but, as the particle becomes more hydrophilic (cos θeq f 1), the two curves diverge substantially. It should be noted that the limiting case of θeq ) 0° (i.e., cos θeq ) 1), which would imply total spreading on a planar substrate, does not necessarily lead to complete coverage on a curved surface such as a particle (or a fiber), especially in the presence of both concave (“peaks”) and convex (“valleys”) regions. On the surface of a rough particle, the liquid preferentially fills the convex regions as can be observed, for example, on the bottom right-hand panel in Figure 8, leaving the concave regions uncovered even if the equilibrium contact angle is close to zero.

4. Conclusions Equilibrium morphologies of liquid droplets on the surface of nonspherical particles have been determined by numerical simulation using the VOF method. The effect of four key parameterssthe particle surface roughness amplitude and correlation length, the equilibrium contact angle, and the relative droplet sizeson the fractional surface coverage as a function of the liquid/solid volume ratio has been systematically investigated. It has been shown that contact angle and relative droplet size influence the fractional surface coverage achievable by a given volume of liquid more strongly than particle shape (as characterized by surface roughness amplitude and correlation length). It has also been observed that, under certain conditions (smaller droplet size and larger values of equilibrium contact angle), increased particle surface roughness can actually lead to higher fractional surface coverage because of the smaller extent of droplet footprint retraction after coalescence. A one-parameter empirical function allowing the estimation of surface coverage as a function of the liquid-to-solid volume ratio, contact angle, droplet size, and particle shape has been

Liquid Droplets on Macroscopically Rough Solids

proposed. The data presented in Figures 10-12 allow the estimation of the particle surface coverage ψ from the liquidto-solid volume ratio xLS using eq 7 for a range of values of particle surface roughness, relative droplet size, and equilibrium contact angle contained in the parametric hypercube [0.2 e a e 0.5] × [0.13 e L e 0.40] × [0.20 e d e 0.63] × [0.25 e cos θeq e 0.99]. The data in this work were presented in a dimensionless form, but there are, of course, limitations on the absolute particle size to which the results are applicable. Neglecting gravity when calculating the equilibrium droplet shape means that there is a limitation from above at approximately 1000 µm. The fact that line tension was not taken into account means that there is a limitation from below at approximately 10 µm. Apart from particle coating, the computational methodology we used can be applied to other situations where the equilibrium distribution of liquid phase on a nonideal surface or in a porous medium is of interest.

Langmuir, Vol. 22, No. 3, 2006 923

f ) volume-of-fluid function h ) spatial discretization step, m L ) surface roughness correlation length (scaling factor rg) Nd ) number of droplets rg ) mean gyration radius of particle, m t′ ) simulation time V ) volume, m3 x ) liquid-to-solid volume ratio Greek Symbols R ) numerical coefficient β ) numerical coefficient  ) convergence criterion θ ) contact angle, rad ψ ) fractional surface coverage Sub- and Superscripts eq ) equilibrium L ) liquid S ) solid

Nomenclature a ) surface roughness amplitude (scaling factor rg) A ) surface area, m2 d ) relative droplet diameter (scaling factor 2rg)

Acknowledgment. Financial support from Merck & Co., Inc. is gratefully acknowledged. LA051901U