# Geometric Interpretation of Surface Tension Equilibrium in Superhydrophobic Systems

^{*}

## Abstract

**:**

## 1. Introduction

^{−2}, while the true value is γ = 0.072 Jm

^{−2}.

## 2. Surface Tension, Surface Free Energy and Contact Angle

#### 2.1. Young Equation

^{−1}) and interfacial energy (measured in joule per square meter, Jm

^{−2}) are often assumed to be identical, they are not exactly the same. The surface tension or, more exactly, the surface stress is the reversible work per unit area needed to elastically stretch a pre-existing surface. The surface stress tensor is defined as ${f}_{ij}=\gamma {\delta}_{ij}+\partial \gamma /\partial {\epsilon}_{ij}$, where ${\epsilon}_{ij}$ is the elastic strain tensor, γ is the surface energy and ${\delta}_{ij}$ is the Kronecker delta. For a symmetric surface, the diagonal components of the surface stress can be calculated as $f=\gamma +\partial \gamma /\partial \epsilon $. For liquids, the interfacial energy does not change when the surface is stretched; however, for solids, $\partial \gamma /\partial \epsilon $ is not zero, because the surface atomic structure of solids are modified in elastic deformation.

_{SL}, γ

_{SV}and γ

_{LV}) and contains a verbal description of Equation (1):

“…for each combination of a solid and a fluid, there is an appropriate angle of contact between the surfaces of the fluid, exposed to the air, and the solid…. We may therefore inquire into the conditions of equilibrium of the three forces acting on the angular particles, one in the direction of the surface of the fluid only, a second in that of the common surface of the solid and fluid, and the third in that of the exposed surface of the solid.” [8]

_{SL}, γ

_{SV}and γ

_{LV}, it does not provide any tool to measure them as an observable quantity, since the only measurable parameter of Equation (1) is the contact angle. Furthermore, the nature of the surface tension has remained a matter of many arguments, including, in particular, how the surface tensions of different phases are related to each other and how these are related to the surface roughness and heterogeneity.

#### 2.2. Antonow’s Equation

_{ab}, is equal to the absolute value of the difference between the individual surface tensions of similar layers, γ

_{a}and γ

_{b}, when exposed to air:

_{max}< γ

_{med}+ γ

_{min}is valid where γ

_{min}< γ

_{med}< γ

_{max}are three interfacial tensions.

_{a}and σ

_{b}[14]. According to Antonoff [14], the words “when exposed to air” were absent from his original definition of the rule [6], and this later addition by other scientist “places the whole principle in contradiction with the laws of thermodynamics.”

#### 2.3. Girifalco and Good’s Equation

#### 2.4. Rough and Heterogeneous Surface

_{0}(Figure 3a), through the non-dimensional surface roughness factor, R

_{f}, equal to the ratio of the surface area to its flat projection

_{1}and the contact angle θ

_{1}and the other with f

_{2}and θ

_{2}, respectively (so that f

_{1}+ f

_{2}= 1), the contact angle is given by the Cassie equation:

_{1}= f

_{SL}, θ

_{1}= θ

_{0}) and liquid to air fraction (f

_{2}= 1 − f

_{SL}, cosθ

_{2}= −1), Equation (7) yields the Cassie–Baxter equation [17]:

#### 2.5. Contact Angle Hysteresis

_{rec}≤ θ ≤ θ

_{adv}, where θ

_{rec}and θ

_{adv}denote the receding and advancing contact angles, respectively. The contact angle can be measured also on a tilted surface (Figure 3c), although it is recognized that the values measured in this way do not always provide true values of the advancing and receding angles [18]. Contact angle hysteresis is small when the solid to liquid adhesion is small, and it is large when the adhesion is large. This makes contact angle hysteresis an important parameter characterizing adhesion, wetting and energy dissipation during the droplet flow.

“The surface tension of a strongly contaminated water surface is variable; that is, it varies with the size of the surface. The minimum of the separating weight attained by diminishing the surface is to the maximum, according to my balance, in the ratio of 52:100. If the surface is further extended, after the maximum tension is attained, the separating weight remains constant, as with oil, spirits of wine, and other normal liquids. It begins, however, to diminish again, directly the partition is pushed back to the point of the scale at which the increase of tension ceased. The water surface can thus exist in two sharply contrasted conditions; the normal condition, in which the displacement of the partition makes no impression on the tension, and the anomalous condition, in which every increase or decrease alters the tension.” [19]

_{0}. These two values are related to the advancing and receding contact angles of the smooth surface, assuming that for a smooth surface, the adhesion hysteresis is the main contributor into the contact angle hysteresis, plus a surface roughness term H

_{r}[23]:

## 3. Graphical Interpretation of the Surface Tension at Equilibrium

#### 3.1. Three-Phase Systems

_{sa}, γ

_{wa}and γ

_{sw}denote the surface tensions for the solid to air, water to air and solid to water interfaces, respectively (Figure 4a.). The surface tensions can be represented as the sides of the triangle with Vertices S, A and W (Figure 4b). Mechanical equilibrium of the droplet requires that the vector sum of the surface tensions be zero, i.e., $\overrightarrow{{\phantom{\rule{0.2em}{0ex}}\mathrm{\gamma}}_{sa}}+\overrightarrow{{\phantom{\rule{0.2em}{0ex}}\mathrm{\gamma}}_{wa}}+\overrightarrow{{\phantom{\rule{0.2em}{0ex}}\mathrm{\gamma}}_{sw}}=0$.

_{max}< γ

_{med}+ γ

_{min}, then the rule receives a clear geometric interpretation of the triangle inequality. The contact angle is a measure of the deviation from the equality of Equation (2).

_{evap}= 45.051 kJ/mol, whereas for ice, the molar enthalpy of melting is h

_{melt}= 6.010 kJ/mol, and the molar enthalpy of sublimation h

_{subl}= 51.059 kJ/mol. Notice that h

_{subl}= h

_{melt}+ h

_{evap}as one would expect. Using the densities ρ

_{ice}= 916.72 kg m

^{−3}and ρ

_{water}= 1000 kg m

^{−3}at 0°C, Lautrup [2] finds γ

_{sa}= 0.138 Jm

^{−2}, γ

_{sw}= 0.016 Jm

^{−2}and γ

_{wa}= 0.129 Jm

^{−2}yielding the value of the contact angle θ = 19° (while the experimentally measured value lies between 12° and 24°).

_{subl}= h

_{melt}+ h

_{evap}, when normalized by water and ice density, yields the triangle inequality, h

_{subl}ρ

_{ice}< h

_{melt}ρ

_{ice}+ h

_{evap}ρ

_{water}, corresponding to the vector triangle $\overrightarrow{{\phantom{\rule{0.2em}{0ex}}\mathrm{\gamma}}_{sa}}+\overrightarrow{{\phantom{\rule{0.2em}{0ex}}\mathrm{\gamma}}_{wa}}+\overrightarrow{{\phantom{\rule{0.2em}{0ex}}\mathrm{\gamma}}_{sw}}=0$ in Figure 4b. This triangle inequality is satisfied only because the density of ice is lower than that of water at 0 °C, ρ

_{ice}< ρ

_{water}, which constitutes the famous “water anomaly.” Thus, the 19° contact angle of water on ice is a measure of the water anomaly. Without the water anomaly, a thin water film would completely cover the ice surface with zero contact angle. Additionally, this is indeed the case for most substances, whose liquid phase wets their solid phase with a thin film, rather than forming a droplet.

#### 3.2. Wenzel and Cassie States in the Three-Phase System

_{f}.

_{1}and S

_{2}, with area fractions f

_{1}and f

_{2}, then the apparent contact angle given by the Cassie–Baxter equation (Equation (7)) can be written as:

_{2}as air, then the equation reduces to $\phantom{\rule{0.2em}{0ex}}\text{cos}{\theta}_{CB}=\phantom{\rule{0.2em}{0ex}}\frac{{f}_{1}(\phantom{\rule{0.2em}{0ex}}{\gamma}_{s1a-}{\gamma}_{s1w})}{{\gamma}_{wa}}-{f}_{2}$.

_{1}and f

_{2}.

#### 3.3. Contact Angle Hysteresis

#### 3.4. Four Phase Systems

^{−2}, 0.072 Jm

^{−2}. A pyramid with surface tension forces can be drawn as shown in Figure 8c.

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) The popular, yet ambiguous description of the origin of surface tension; the imbalance of cohesive forces between molecules due to the lack of bonds at the surface results in surface tension. (

**b**) The network of tetrahedral water molecules at the air-water interface is more ordered than that in the bulk. Consequently, the decrease of entropy TΔS due to the additional orderliness in the surface layer may be partially responsible for the surface tension. (

**c**) When the interface between A and B is displaced along the vector $\phantom{\rule{0.2em}{0ex}}\overrightarrow{dr}$, the surface tension force acts on the three-phase line l in the direction of the normal $\phantom{\rule{0.2em}{0ex}}\overrightarrow{n}$.

**Figure 2.**(

**a**) Surface tension forces at the three-phase line on a non-deformable solid surface. The vertical components of the forces remain unbalanced. (

**b**) The equilibrium of surface tensions at the three-phase line of liquid Phases A, B and C. Note the deformation of the interface between A and C.

**Figure 3.**(

**a**) A liquid droplet in the Wenzel state, with a homogenous solid to liquid interface below the droplet; (

**b**) a liquid droplet in the Cassie–Baxter state, with a composite solid to liquid to vapor interface below the droplet; (

**c**) contact angle hysteresis (CAH) measurement by tilting the droplet. The maximum or advancing (θ

_{adv}) and minimum or receding (θ

_{rec}) contact angles are measured at the front and rear of a moving droplet, respectively.

**Figure 4.**(

**a**) Equilibrium of surface tension vectors at the three-phase line; (

**b**) Neumann’s triangle for a three-phase system; (

**c**) when Antonoff’s rule is an exact equality, the surface tension vectors lie on the same line; this corresponds to complete wetting; (

**d**) geometric interpretation of the Girifalco and Good equation.

**Figure 6.**(

**a**,

**b**) Neumann’s triangles for three-phase systems S1AW and S2AW, respectively; (

**c**) Neumann’s triangle for a three-phase Cassie–Baxter state.

**Figure 7.**(

**a**) Contact angle hysteresis represented using Neumann’s triangle as a rotation of the vector $\overrightarrow{{\gamma}_{wa}}$. A range of contact angles are possible under the constraint that $|\phantom{\rule{0.2em}{0ex}}\overrightarrow{{\gamma}_{wa}}|$ remains constant. (

**b**) The forces involved in contact angle hysteresis. For a droplet on a tilted surface, the pressure inside acting normal to the surface is the sum of Laplace, as well as hydrostatic pressures. (

**c**) The components of the surface tension vectors normal to the surface balance the pressure force, while the components of the surface tension vectors along the surface balance the friction force.

**Figure 8.**(

**a**) Tetrahedron of surface tension vectors in 3D space for a four-phase system; (

**b**) a four-phase system on aluminum-graphite composite, which consists of a vegetable oil (γ

_{o}= 0.032 Jm

^{−2}) droplet (volume about 5 μL) on Al-C composite immersed in water (γ

_{w}= 0.072 Jm

^{−2}) with pockets of air trapped on the Al-C surface forming the fourth phase; (

**c**) the tetrahedron of surface tension vectors for the four-phase system.

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Nosonovsky, M.; Ramachandran, R.
Geometric Interpretation of Surface Tension Equilibrium in Superhydrophobic Systems. *Entropy* **2015**, *17*, 4684-4700.
https://doi.org/10.3390/e17074684

**AMA Style**

Nosonovsky M, Ramachandran R.
Geometric Interpretation of Surface Tension Equilibrium in Superhydrophobic Systems. *Entropy*. 2015; 17(7):4684-4700.
https://doi.org/10.3390/e17074684

**Chicago/Turabian Style**

Nosonovsky, Michael, and Rahul Ramachandran.
2015. "Geometric Interpretation of Surface Tension Equilibrium in Superhydrophobic Systems" *Entropy* 17, no. 7: 4684-4700.
https://doi.org/10.3390/e17074684