Capillary force approach

Due to fabrication of needle-like surfaces through an electrochemical etching process in an aqueous HF solution and cleaning of the surfaces with deionized water, needles’ surfaces are hydrogen terminated. These surfaces are not stable at ambient atmosphere and oxidize immediately [254]. Thus, due to the hydrophilic nature of oxidized silicon surfaces, these surfaces can adsorb several monolayers of water in normal humid air, which can promote formation of meniscuses and associated capillary bridges and forces upon their proximities and contacts [255]. These attractive capillary forces remain and dominate over VdW forces at any distance beyond the interatomic spacing as long as the meniscuses are formed [256–258].

In general, when two bodies are brought into a contact, a small gap around their

contact area is created. When the environment is humid or surfaces are lyophilic with respect to surrounding vapors, some vapor condenses into the liquid phase and form a meniscus with surfaces if the provided gap between them is sufficiently small [259].

The meniscus causes an attractive force (capillary forces) and pulls the bodies towards each other, resulting from the direct interaction of the surface tension of the liquid around periphery of the meniscus [260]. The capillary forces are reduced by increasing the gap between two bodies and present as long as the meniscus remains [261].

The needle-substrate interaction of a needle in the capillary force approach can be obtained through the interaction between its hemispherical head and the substrate similar to the contact mechanics approach since the condensed vapor creates only

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meniscus between the hemispherical head of the needle and the opposing substrate.

Hence, the sphere-plane capillary force model [126, 131, 138, 140, 141, 259] can be used to express the capillary needle-substrate interaction forces (𝐹_{𝑁−𝐶𝑎𝑝}). When a hemispherical body with radius (𝑅)stays at the distance (𝑑) from a plane, the enclosed liquid between the hemisphere and the plane forms a meniscus with contact angles 𝜃1 and 𝜃2 with the sphere and the plane, respectively (see Figure 52). Curvature of the meniscus, also called pendular ring, can be characterized by two radii: the azimuthal radius (𝑙_𝑟) and the meridional radius (𝑟_𝑟) in which the meridional radius is perpendicular to the azimuthal radius. In this way, the azimuthal radius is considered positive since it is concave with respect to the liquid, and the meridional radius is considered negative.

Figure 52. 2D schematic of the hemisphere-plane capillary interaction and its geometry parameters (Retrieved from[126]).

Assuming a circle with radius 𝑟𝑟 for the curvature of the meniscus (circular or toroidal approximation), the Laplace pressure inside the liquid can be expressed as [201]:

∆𝑃 = 𝛾 (1 𝑙𝑟− 1

𝑟𝑟) (55) This pressure is lower than the pressure in outer vapor phase. It acts upon a cross-sectional area 𝜋𝑙_𝑟² and leads to an attractive force of 𝜋𝑙_𝑟²∆𝑃. Hence, the total capillary forces between the hemisphere and the plane (𝐹𝑁−𝐶𝑎𝑝) can be obtained by combining

125 the force generated by the pressure difference across the meniscus with the force generated by the surface tension acting along the meniscus as [126]:

𝐹_{𝑁−𝐶𝑎𝑝} = 2𝜋𝛾𝑙_𝑟+ 𝜋𝑙_𝑟²∆𝑃 (56) By introducing a parameter, the so-called filling angle (β) to describe the position of the three-phase contact line, height of the meniscus can be described as:

ℎ = 2𝑟𝑟𝑐 = 𝑅(1 − cos (𝛽)) + 𝑑 (57) with

𝑐 =cos(𝜃₁+ 𝛽) + cos(𝜃₂)

2 (58) Using Eq. (57), Eq. (58), and geometrical relations between the parameters, the meridional and the azimuthal radii of curvature of the meniscus can be expressed as:

𝑟𝑟 = 𝑅(1 − cos ( 𝛽)) + 𝑑

cos(𝜃1+ 𝛽) + cos ( 𝜃2) (59) 𝑙_𝑟 = 𝑅 sin(𝛽) − 𝑟_𝑟[1 − sin(𝜃₁ + 𝛽)] (60) By inserting Eq. (55), Eq. (59), and Eq. (60) into Eq. (56), and considering the force at three-phase contact line, where the surface tension component is not vertically directed and its horizontal component (sin(𝜃1+ 𝛽)) contributes only, the total capillary forces (𝐹𝑁−𝐶𝑎𝑝) can be written as [259]:

𝐹𝑁−𝐶𝑎𝑝 = 𝜋𝛾𝑅 sin(𝛽) [2 sin(𝜃1+ 𝛽) + 𝑅 sin ( 𝛽) (1 𝑟𝑟− 1

𝑙𝑟)] (61) This equation is not an explicit equation with respect to the vapor pressure since capillary forces cannot be simply calculated by inserting the vapor pressure. However, to calculate capillary forces for a given hemisphere size, contact angles (𝜃₁ and 𝜃₂), and vapor pressure, parameters 𝑟𝑟 and 𝑙𝑟 have to be calculated according to β.

Assuming 𝑅 ≫ 𝑙_𝑟 ≫ 𝑟_𝑟, an explicit expression for Eq. (61) in term of radius of curvature of the meniscus can be written as [126]:

𝐹_{𝑁−𝐶𝑎𝑝}≈ 2𝜋𝛾𝑅²(1 − cos (𝛽))

𝑟_𝑟 (62) where 𝑟_𝑟 = 𝑟_𝑐 and it can be simply calculated by the Kelvin equation (Eq. 22) for a given relative humidity.

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The capillary force interaction between interlaced needles (the needle-needle interaction) can be simplified to the interaction between two parallel adjacent needles from opposite surfaces (see Figure 53). Assuming a needle as a vertical cylinder with a hemispherical head, the capillary force model for the cylinder-plane interaction proposed by Vitard et al. [235] can be used to describe the needle-needle interaction with a slight modification. The same configuration as is shown in Figure 52, can be also used to describe the cylinder-plane interaction. In this case, R is radius of the vertical cylinder, and the 𝜃₁ and 𝜃₂ are contact angles between water and the cylinder and water and the plane; respectively.

Figure 53. Geometrical parameters of interlaced needles in the capillary needle-needle interaction.

The capillary force between the cylinder and the plane (𝐹_{𝐶𝑝−𝐶𝑎𝑝}) can be simply expressed as sum of the Laplace pressure difference and the surface tension term as [235]:

𝐹_{𝐶𝑝−𝐶𝑎𝑝} = 2𝑙_𝑟𝐿∆𝑃 + 2𝐿𝛾 sin(𝜃₂) (63) Inserting Eq. (55), Eq. (59), and Eq. (60) into Eq. (63), and using geometrical relations, the 𝐹𝐶𝑝−𝐶𝑎𝑃 can be expressed as:

𝐹𝐶𝑝−𝐶𝑎𝑃 = 2𝐿𝛾 (𝑅 sin(𝛽)cos(𝜃2) + cos( ß + 𝜃2)

𝑑^′+ 𝑅(1 − cos(𝛽)) + sin(𝜃1+ 𝛽)) (64) where L is length of the cylinder

127 Substituting R with equivalent radius (𝑅_𝑒𝑞 =_𝑅^2𝑅1𝑅2

1+𝑅₂ ) [127, 143, 262] and 𝐿 with overlapping length of interlaced needles (𝐿_𝐶 = 𝐿₁+ 𝐿₂− 𝐷) and using Eq. (59), Eq.

(64) can be modified to express the capillary interaction forces between two interlaced needles (𝐹_{𝑆−𝐶𝑎𝑝}) with different lengths (𝐿₁and 𝐿₂) in term of radius of curvature of the meniscus (𝑟𝑟) defined by the Kelvin equation as:

𝐹_{𝑆−𝐶𝑎𝑝} = 2𝐿_𝐶𝛾 (𝑅_𝑒𝑞 𝑠𝑖𝑛𝛽

𝑟_𝑟 + sin(𝜃₁+ 𝛽)) (65)