Producing super-hydrophobic surfaces with nano-silica spheres

Rob J. Klein; P. Maarten Biesheuvel; Ben C. Yu; Carl D. Meinhart; Fred F. Lange

doi:10.1515/ijmr-2003-0068

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Producing super-hydrophobic surfaces with nano-silica spheres

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Veröffentlicht/Copyright: 3. Februar 2022

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Aus der Zeitschrift International Journal of Materials Research Band 94 Heft 4

Abstract

Polycrystalline alumina substrates were dip-coated in dilute suspensions formed with dispersed, nano-silica particles. The fractional surface coverage of the alumina substrate was varied between ≈ 0.05 to ≈ 0.4 by changing concentration of particles in the silica slurry. After a heat treatment to partially sinter the particles to the surface, the surface was made hydrophobic by a reaction with a solution containing fluorosilane molecules. Wetting measurements showed that the contact angle between the surface and water droplets increased with decreasing area coverage of nano-silica spheres, consistent with a previous theory, modified here to include the pressure of the water droplet. The modification of the theory predicts that the super-hydrophobic effect disappears when the particles become too widely spaced, causing the pressure of the water droplet to spontaneously wet the substrate.

Keywords: Nano-silica spheres; Super-hydrophobic surfaces

Dedicated to Professor Dr. Dr. h. c. Manfred Rühle on the occasion of his 65th birthday

Professor Fred F. Lange Materials Dept., UCSB, Eng. III Santa Barbara, CA 93106-5080, USA Tel.: +1 805 893 8248 Fax: +1 805 893 8486

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Appendix: Conditions for Spontaneous Wetting

We assume that the breakdown of the super hydrophobicity effect results from one of two phenomena. Both are caused by the finite weight of the droplet causing the meniscus to have a finite curvature. The first mechanism occurs when the pressure differential, P, across the meniscus becomes greater than a critical value, P_cr. When P = P_cr, the liquid will no longer wets the tops of the spheres, but the meniscus will spontaneously move to the substrate, thus wetting most of the spheres and the substrate. The second mechanism occurs when center of the curved meniscus touches the substrate, which will cause the shape of the meniscus to change and partially wet the substrate.

Let us begin by idealizing the randomly distributed particles, of diameter 2r, as a periodic, square array with spacing 2l, as shown in Fig. A – 1. Relative to the atmospheric pressure, the liquid will expert a pressure forcing the meniscus between the spheres and thus forcing it to have a finite curvature. Fig. A – 2 depicts the geometry of the liquid meniscus and its contact with the particles. The contact angle is denoted by θ, while β is angle between the tangent of the particle surface at the point of contact and the horizon, and α is the angle between the air/liquid interface and the horizon.

One can shown that

(A-1) α=θ+β−π

φ is the projected area fraction of the spheres on the surface expressed as

(A-2) φ=πr24l2

And d, the diameter of the wetted area of each sphere is given by

(A-3) d=2rsinβ

As first taught be Laplace, for a given differential pressure, P, the equilibrium position of the meniscus can be determined by equating the vertical (y direction) force where the spheres support the meniscus to the opposing force exerted by the differential pressure on the meniscus. At equilibrium, the sum of these forces are

(A-4) ∑Fy=πdγsinα−P(4l2−πd24)=0

where γ is the surface energy per unit area of the meniscus.

Fig. A1

. Periodic Spheres on Surface

Using the above equations, one can show that

(A-5) P=−2φγsinβsin(θ+β)r(1−φsin2β)

Equation (A–4) is equivalent to the Laplace equation, relating the differential pressure, P, across the meniscus to the geometry, defined by r, φ and β, and to the specific surface energy (γ) and wetting angle (θ).

Inspection of Eq. (A–4) shows that the maximum pressure the meniscus can sustain without being forced to the substrate occurs when β = π/2. Thus, the largest and thus critical pressure the meniscus can support is given by

(A-6) Pc=−2φγcosθr(1−φ) or [ ≈−2φγcosθr, when φ<0.1 ]

Eq. (A–5) shows that, for a given size droplet, spontaneous wetting will occur below a critical volume fraction, i. e., the wetted particles become too widely spaced to support the meniscus. This relation also shows that the critical pressure for spontaneous wetting is inversely related to particle size.

The force exerted by a drop of diameter D is given by

(A-7) F=π6gρD3

where g is the acceleration of gravity and ρ is the density of the liquid. This force acts on a contact area (0.25 π D² sin² θ*) to produce a pressure on the meniscus that can be estimated, for θ > 135°, to be

(A-8) Pd=2gDρ3sin2θ∗

One determines the critical area fraction for spontaneous wetting by equating Eqs. (A– 6) and (A– 8)

(A-9) φc=−gDρr3γcosθsin2θ∗

To test the second mechanism, the equilibrium shape of the meniscus was determined using the finite element package FEMLab^TM (http://www.comsol.com). For the range of parameters of interest here, the results of the simulation showed that the contact line reaches the critical point on the particles (i. e. β = π/2 ) before the center of the meniscus touches the substrate.

Fig. A2.

Meniscus between spheres

Received: 2003-01-10

Published Online: 2022-02-03

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Artikel in diesem Heft

https://doi.org/10.1515/ijmr-2003-0068

Schlagwörter für diesen Artikel

Nano-silica spheres; Super-hydrophobic surfaces