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A Cassie–Baxter state is a composite wetting state in which a drop contacts solid asperities over area fraction $f_s$ and trapped gas over the remainder. An ideal area-weighted contact-angle model gives the displayed apparent angle from the smooth-solid Young's equation angle ${\theta }_Y$. Reducing solid contact can strongly enhance apparent hydrophobicity. Such a state can be metastable and can collapse when liquid enters the texture; roughness alone does not guarantee it.

A contact line can remain pinned while the observed contact angle changes between receding and advancing limits. Surface roughness and chemical heterogeneity can produce this hysteresis. Slowly adding or removing liquid from a sessile drop measures advancing and receding angles separately. An observed pinned angle therefore need not be the single equilibrium angle predicted by Young's equation for a perfectly homogeneous surface.

Subtracting Young's equations for two liquids on the same solid eliminates the unknown solid–vapour interfacial energy. The result assumes equilibrium angles and an unchanged solid surface. Swelling, contamination or contact-angle hysteresis can invalidate this cancellation as a comparison of equilibrium material properties.

Use unambiguous symbols ${\gamma }_{SV},{\gamma }_{SL},{\gamma }_{LV}$ for solid–vapor, solid–liquid and liquid–vapor interfacial energies. The printed description repeats an interface subscript, so it cannot be used literally for two different interfaces.

For a smooth homogeneous equilibrium surface, take a horizontal displacement of the contact line. The substrate-parallel force balance isThus the contact-angle relation is Young's equation:The same result follows by varying interfacial free energy at fixed liquid volume. It assumes equilibrium, no contact-line pinning, and negligible line-tension and roughness corrections. If the inferred cosine lies outside $[-1,1]$, the finite-angle equilibrium construction fails and a wetting limit must be considered.

The name in the question should be distinguished from the Young–Laplace equation, which relates a pressure jump to interface curvature. The horizontal interfacial balance derived here is Young's equation; it does not require introducing a pressure-curvature formula.

This is the first printed “(vi)”, concerning adhesion. The reversible work per unit area to separate the solid–liquid interface into solid–vapor and liquid–vapor interfaces isSubstitute ${\gamma }_{SV}-{\gamma }_{SL}={\gamma }_{LV}\cos {\theta }_C$ from Young's equation:This is the Young-Dupré equation. It has units $\mathrm{J}{\mathrm{m}}^{-2}$, equivalent to $\mathrm{N}{\mathrm{m}}^{-1}$. The liquid–vapor tension is the multiplier; the abbreviated printed subscript does not denote a new interface. Within this equilibrium model, smaller water contact angle means larger reversible work of adhesion for fixed liquid–vapor tension.

Advance the projected contact line by $dx$ per unit transverse length without wetting the gaps. A fraction $f$ converts solid–vapor interface into solid–liquid interface, contributing $f({\gamma }_{SL}-{\gamma }_{SV})dx$. The remaining fraction creates liquid–vapor interface above trapped gas, costing $(1-f){\gamma }_{LV}dx$. The outer droplet surface contributes ${\gamma }_{LV}\cos {\theta }^{\ast }dx$. ThusSubstituting Young's equation givesThis is the ideal Cassie–Baxter state relation. Its $f$ is projected solid-contact fraction, not the roughness factor of Wenzel wetting. A suspended interface over the gaps is essential; invasion of the texture changes the area bookkeeping.

At a straight contact line, take forces per unit transverse length and measure the contact angle $\theta$ through the liquid. Solid–vapor surface tension pulls outward along the solid, solid–liquid surface tension pulls inward, and the liquid–vapor force has horizontal component ${\gamma }_{LV}\cos \theta$. Tangential equilibrium givesThis is Young's equation. The substrate supplies the normal reaction balancing the vertical capillary component. The original force sketch labels directions at a right-hand contact line. The result assumes smooth, chemically uniform, rigid surfaces in partial-wetting equilibrium, with negligible pinning and line tension.

Wetting concerns how a liquid covers a solid in contact with another phase, such as its vapor. For an ideal flat homogeneous surface, balance of the three interfacial tensions gives Young's equationwhere the contact angle $\theta$ is measured through the liquid. Small angle means favorable contact; partial wetting gives a finite angle. The spreading coefficient $S={\gamma }_{SV}-{\gamma }_{SL}-{\gamma }_{LV}$ compares a covered surface with a dry one. Positive $S$ favors complete wetting, while negative $S$ permits a finite-angle droplet in the partial-wetting regime.

A sessile-drop experiment images the tangent at the contact line on a clean level substrate. Varying drop volume slowly distinguishes advancing and receding angles; their difference is contact-angle hysteresis from pinning and heterogeneity, so a single observed angle need not equal the ideal equilibrium value. A capillary-rise experiment gives a second consequence: the Young–Laplace equation and hydrostatic balance predict $h=2{\gamma }_{LV}\cos \theta /(\rho gr)$ in a narrow circular tube when meniscus-volume corrections are negligible. A wetting liquid rises, whereas a sufficiently nonwetting one is depressed.

For a smooth homogeneous solid and a partially wetting liquid at equilibrium, tangential balance at the contact line gives Young's equationThe angle is measured through the liquid. Replacing solid-vapour area by solid-liquid plus liquid-vapour area changes free energy by $-S$ per wetted area, where the spreading coefficient isFor partial wetting, $S={\gamma }_{LV}(\cos \theta -1)<0$. Positive $S$ favours a spreading film and zero macroscopic equilibrium contact angle; the finite-angle Young construction cannot be extrapolated to $\cos \theta >1$.

The intended chemically matched surface approximation is ${\gamma }_{SL}\simeq 0$. It gives $S\simeq 23-21.3=1.7\mathrm{mN}{\mathrm{m}}^{-1}>0$, so increasing the wetted area lowers free energy. The initially steep drop spreads and flattens toward complete wetting:The initial $90^{\circ }$ angle describes a nonequilibrium shape and must not be inserted into equilibrium Young's equation to fix ${\gamma }_{SL}$. More precisely, complete wetting requires ${\gamma }_{SL}\le 1.7\mathrm{mN}{\mathrm{m}}^{-1}$ in this surface-energy model; chemical similarity alone is not a measurement of that interfacial energy.

Water on such a low-energy, oil-like surface is expected to bead into a comparatively rounded, high-angle drop because the substrate does not compensate its cohesive interactions. Its numerical contact angle needs the water-solid interfacial energy, which is not supplied.

Young's equation gives ${\gamma }_{SV}-{\gamma }_{SL}={\gamma }_{LV}\cos \theta$. The interfacial-energy difference from two contact angles follows by subtracting the two equilibrium contact-angle equations, which eliminates the common solid–air energy:The water–solid interface has the larger energy. This assumes the same clean solid surface and equilibrium angles, without swelling or strong contact-angle hysteresis.

For an equilibrium contact angle measured through the liquid, Young's equation balances the tangential interfacial forces:The pure-ethanol measurement therefore fixes the solid–vapour surface energy asThe inferred surface energy is $40.7\mathrm{mJ}{\mathrm{m}}^{-2}$, equivalently $40.7\mathrm{mN}{\mathrm{m}}^{-1}$. The equality of these units follows because creating interfacial area costs energy.

For the equal-fraction mixture, the stated linear relations give ${\gamma }_{SL}=20.5\mathrm{mN}{\mathrm{m}}^{-1}$ and ${\gamma }_{LV}=28.5\mathrm{mN}{\mathrm{m}}^{-1}$. Holding the solid–vapour surface energy fixed,The small decrease in ${\gamma }_{SL}$ increases the numerator of Young's equation, but the increase in ${\gamma }_{LV}$ has the larger relative effect and reduces $\cos \theta$. Consequently the contact angle increases and the mixture wets the surface less well. This inference uses equilibrium wetting on an unchanged smooth solid; a changing surface composition or contact-angle hysteresis would require additional measurements.

Molecular area from the adsorption slope. The supplied Gibbs adsorption equation identifies the surface excess number density $\Gamma =1/\Sigma$ throughThe logarithm is understood relative to a fixed concentration unit or reference concentration; only concentration ratios enter its differences. The pure-water point $c=0$ supplies a baseline surface tension, but cannot be included in $\ln c$.

An estimate near the later working concentration $0.055\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$ uses its two bracketing points:Taking room temperature as $298\mathrm{K}$,This corresponds to about $4.4$ adsorbed molecules per $\mathrm{n}{\mathrm{m}}^2$. The data are not perfectly linear in $\ln c$, so the inferred molecular area depends somewhat on the concentration range: a fit to all five positive-concentration points gives about $0.26\mathrm{n}{\mathrm{m}}^2$, while the higher-concentration region gives roughly $0.22\mathrm{n}{\mathrm{m}}^2$. An effective area of order $0.2$–$0.3\mathrm{n}{\mathrm{m}}^2$ is justified; a concentration-independent exact value is not.

Contact angle. Linear interpolation between $0.05$ and $0.06\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$ gives ${\gamma }_{LV}(0.055)=(48.5+45.2)/2=46.85\mathrm{mN}{\mathrm{m}}^{-1}$. For an ideal equilibrium contact angle measured through the liquid, Young's equation isThe supplied solid-vapour and solid-liquid values have difference $40\mathrm{mN}{\mathrm{m}}^{-1}$. HenceThis assumes clean homogeneous equilibrium interfaces, negligible contact-angle hysteresis, and the stated solid interfacial energies applicable to the solution.

Complete wetting. The spreading coefficient is $S={\gamma }_{SV}-{\gamma }_{SL}-{\gamma }_{LV}=40\mathrm{mN}{\mathrm{m}}^{-1}-{\gamma }_{LV}$. The onset of complete wetting occurs at $S=0$, or ${\gamma }_{LV}=40\mathrm{mN}{\mathrm{m}}^{-1}$, if the solid interfacial energies remain unchanged. The last measured value is still $42.0\mathrm{mN}{\mathrm{m}}^{-1}$, so this requires extrapolation beyond the table.

Continuing the last local slope in concentration, $(42.0-45.2)/(0.07-0.06)=-320\mathrm{mN}{\mathrm{m}}^{-1}$ per $\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$, givesContinuing a constant local Gibbs slope in $\ln c$ instead gives approximately $0.078\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$. Thus the data suggest a threshold near $0.08\mathrm{M}$ under continued-trend assumptions, rather than fixing one exactly. A surface-tension plateau caused by micellization, or a change in ${\gamma }_{SL}$, could prevent or shift this threshold; no such extra data are given.

Capillary rise. For a circular capillary of radius $a$, vertical surface-tension force $2\pi a{\gamma }_{LV}\cos \theta$ balances the weight $\rho g\pi a^{2}h$. Thus capillary action givesUsing $a=0.5\mathrm{mm}=5.0\times 10^{-4}\mathrm{m}$, $\rho =10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$,This is the height above the reservoir level in the ideal static capillary approximation; meniscus-volume corrections and contact-angle hysteresis are neglected. The positive $\cos \theta$ gives a rise, rather than a depression.

A hydrophobic surface is poorly wetted by water: on a smooth surface its equilibrium contact angle, measured through the liquid, exceeds $90^{\circ }$. A drop therefore tends to bead up, limiting its contact area. Hydrophobicity does not mean that the surface exerts a mysterious repulsive force on every water molecule.

The waxy coating has a largely nonpolar exterior. Water molecules form favorable hydrogen bonds with one another, whereas a hydrocarbon-rich wax surface offers few suitable hydrogen bond donors or hydrogen bond acceptors. Water–surface adhesion is weak relative to water's cohesion. The drop shape minimizes the combined solid–air, solid–water and water–air interfacial free energies, not merely the liquid's exposed area.

In terms of Young's equation, ${\gamma }_{SV}-{\gamma }_{SL}={\gamma }_{LV}\cos \theta$. A contact angle greater than $90^{\circ }$ corresponds to ${\gamma }_{SL}>{\gamma }_{SV}$: replacing dry surface by wetted surface has an interfacial cost. The balancing liquid interface then favors a small wetted patch. Microscopic leaf texture can amplify the apparent hydrophobicity, especially when it traps air under the drop as a Cassie–Baxter state; the chemistry of a smooth wax layer alone does not guarantee an extremely large apparent angle.

For an ideal smooth homogeneous substrate in thermodynamic equilibrium, Young's equation iswhere $A$ denotes air, $S$ the solid, and $L$ the chosen liquid. It follows either by balancing tangential interfacial forces at the contact line or by varying the wetted area at fixed drop volume.

The dry Teflon–air surface energy is the same in the two measurements, so subtracting the two equations eliminates it:ThereforeThe water–Teflon interface has the higher interfacial free energy per unit area. Neither individual solid–liquid surface energy is determined without the solid–air value; the difference is determined.

The excess interfacial free energy released per newly wetted area when a liquid film replaces a solid-vapour interface is the displayed coefficient. Positive $S$ favors macroscopic spreading in the ideal equilibrium surface-energy model, whereas negative $S$ permits a finite contact angle obeying Young's equation. Surface preparation, chemical composition and adsorption can change all three interfacial energies.

Combining reversible work of adhesion per area, $W={\gamma }_{SV}+{\gamma }_{LV}-{\gamma }_{SL}$, with Young's equation gives the displayed relation. It assumes the same smooth equilibrium interfaces; contact-angle hysteresis and dissipation are not included.

For a smooth homogeneous solid in thermodynamic equilibrium with a liquid and vapor, Young's equation relates their three interfacial tensions to the contact angle measured through the liquid. A virtual increase $dA$ of wetted area changes the interfacial free energy by $({\gamma }_{SL}-{\gamma }_{SV}+{\gamma }_{LV}\cos \theta )dA$. Setting this to zero proves the equation. Roughness, pinning and chemical heterogeneity can make an observed apparent angle differ from this ideal equilibrium value. The equation does not determine each solid–liquid surface energy from a contact angle alone, but two liquids on the same unchanged dry substrate allow the solid–vapor term to be eliminated by subtraction.