According to Young a contact angle is defined as the angle *θ*_{Y} formed by a liquid such as water on an ideal solid at the three phase boundary, where liquid (*L*), vapour (*V*) and solid (*S*) forming the interfacial tensions (*σ*) of the contacting boundaries at equilibrium according to the equation:

$$\mathrm{cos}\hspace{0.17em}{\theta}_{Y}=\frac{{\sigma}_{S{V}_{Y}}-{\sigma}_{S{L}_{Y}}}{{\sigma}_{L{V}_{Y}}}$$(1)

Recently we described contact angles as complex numbers consisting of a real part (Young contact angle) and an imaginary part (imaginary contact angle) [2]:

$$\mathrm{\Theta}=\left\{{\theta}_{\text{Y}}\right\}+\left\{{\lambda}_{i}\right\}\mathit{\hspace{1em}\hspace{1em}}{[}^{\circ}]$$(2)

where Θ is the observed (effective) complex contact angle {*θ*_{Y}} the real or Young contact angle and {*λ*} the imaginary part together with the imaginary unit i (imaginary contact angle, [2]). Experimentally the intrinsic contact angle {*θ*_{0}} is defined as {*θ*_{0}} ∼ {*θ*_{Y}} ([2]; see below).

The *second approach* to complex contact angles [6] is based on the reports of Xiong et al. [7], [8], which indicate that the surface tension of water (${\sigma}_{LV}^{*}$) is a complex quantity consisting of a real and an imaginary part in the form of ${\sigma}_{LV}^{*}$ = 73 + 17i mN/m at 26 °C [7].

Assuming this is correct, the Young equation can be rewritten as a complex number: Insertion of the complex surface tension into the Young equation (eq. 1) converts the Young equation to a complex contact angle equation:

$$\mathrm{cos}\hspace{0.17em}\mathrm{\Theta}=\frac{{\sigma}_{S{V}_{0}}-{\sigma}_{S{L}_{0}}}{({\sigma}_{L{V}_{0}}+\mathit{\hspace{1em}}\sigma {}^{\mathrm{\prime \prime}}{}_{L{V}_{C}}i)}$$(3)

Where Θ is a complex contact angle, the terms ${\sigma}_{S{V}_{0}}$ (solid-vapor surface tension), ${\sigma}_{S{L}_{0}}$ (solid liquid interfacial tension) and ${\sigma}_{L{V}_{0}}$ (73 mN/m) i.e. the real part of the surface tension carries the subscript 0 indicating intrinsic conditions (e.g. the intrinsic contact angle). The term $\sigma _{L{V}_{\u2102}}{}^{\mathrm{\prime \prime}}$ with the subscript denotes the imaginary component of the complex surface tension. If $\sigma {}^{\mathrm{\prime \prime}}{}_{L{V}_{\u2102}}=0$, then eq. 3 reduces to eq. 1, i.e. the classical Young equation [2].

The complex surface tension of water can also be inserted into the buoyancy abridged Wilhelmy equation, which for ideal conditions has the following form:

$$\mathrm{cos}\mathit{\hspace{1em}}{\theta}_{Y}=\frac{F}{P\times {\sigma}_{L{V}_{Y}}}$$(4)

where *F* is the measured net force, *P* is the perimeter of the sample, ${\sigma}_{L{V}_{Y}}$ the classical i.e. Young surface tension of water and *θ*_{Y} the dynamic Young contact angle with absent hysteresis.

Substituting the complex surface tension into eq. 4 we obtain the Wilhelmy equation for complex contact angles:

$$\mathrm{cos}\mathit{\hspace{1em}}\Theta =\frac{F}{P}\left(\frac{1}{{\sigma}_{L{V}_{\mathit{0}}}+\mathit{\hspace{1em}}\sigma {}^{\mathrm{\prime \prime}}{}_{L{V}_{\mathrm{C}}}i}\right)$$(5)

which after rearrangements leads to two equations, the first of which is a complex trigonometric number according to:

$$\mathrm{cos}\hspace{0.17em}\mathrm{\Theta}=A\u2022F\left({\sigma}_{L{V}_{0}}-\sigma {}^{\mathrm{\prime \prime}}{}_{L{V}_{\u2102}}i\right)$$(6)

With the constant $A={\left[P({\sigma}_{L{V}_{0}}^{2}+\sigma {}^{\mathrm{\prime \prime}}{}_{L{V}_{\mathrm{C}}}{}^{2})\right]}^{-1}$. In equation 6 the cosine of the contact angle is directly proportional to the force of the Wilhelmy balance and the complex surface tension of water. The second resulting equation is:

$$\mathrm{sec}\mathit{\hspace{1em}}\mathrm{\Theta}=\frac{P}{F}\left({\sigma}_{L{V}_{0}}+\sigma {}^{\mathrm{\prime \prime}}{}_{L{V}_{\u2102}}i\right)$$(7)

Thus the secant of the complex contact angle Θ is equal to the product of the ratio P/F and the complex surface tension also leading to a complex trigonometric number.

The imaginary part of complex contact angles was however discovered in the *first approach* in a more direct manner and may indicate a different origin of complex contact angles. It was found that measurements made on highly hydrophilic rough titanium surfaces with the Wilhelmy balance constantly led to force values leading to cos *θ* > 1 [1], which according to current dogma are undefined. In reality however, the solution to the inequality cos *θ* > 1 is the imaginary part of a complex contact angle [1]. In order to account for the additional forces exerted by rough surfaces in the Wilhelmy balance, an extended Wilhelmy equation was suggested [11]:

$$\mathrm{cos}\hspace{0.17em}\mathrm{\Theta}=\frac{{F}_{W}+\kappa {F}_{\mathrm{\Sigma}}-{F}_{\mathrm{Imb}}}{P\u2022\gamma}$$(8)

where *F*_{W} is the Wilhlemy force and *F*_{Σ} is the sum of all additional forces acting on the miniplate that are not accounted for by the Young equation and *κ* is a constant. *F*_{Imb} is the force exerted by the impregnation or imbibition of the rough microstructure with water, which can be corrected for by baseline correction. This is the equation with a wide applicability to the analysis and study of surfaces of a given solid *when not smooth* according to the definition of Dettre et al. 1967 [2], [13]. Examples for the application of this equation to rough titanium surfaces of dental implants are given in the reports [3], [4].

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