## 1 Introduction

Historically there have been two approaches to complex contact angles. In the *first approach* of 2011 [1] we reported that Wilhelmy balance data of rough surfaces in the ultra-/superhydrophilic range often cause forces leading to values of cos *θ* > 1, which at that time had been termed as undefined and erroneous. We showed that the solution of this inequality is an imaginary number, which we interpreted as a dynamic imaginary contact angle, for the case that the real part of the underlying complex contact angle (i.e. intrinsic contact angle) is zero [2]. Concerning the question of a “physical basis”, we felt that such a basis, which may as well be a “chemical basis”, is important but not mandatory for an applied mathematical necessity (see [2]). This approach immediately enabled a novel analysis of hyperhydrophilic surfaces on dental implants [3], [4]. It should be recalled that according to current doctrine all ultra-/superhydrophilic surfaces by definition have the peculiarity of possessing the same lowest contact angle of zero degrees. According to our view this is incorrect, since imaginary or complex contact angles can be assigned to many of these surfaces, which we have termed “hyperhydrophilic” to distinguish them from ultra-/superhydrophilic surfaces [5]. In the *second approach* of 2015 [6] we showed that in principle it is also possible to calculate complex contact angles from the complex surface tension of water [7]. The complex surface tension of water [7], [8] remains to be further investigated. On the other hand the second approach enables a calculation of the surface tension from dynamic complex contact angles. Thus, for the experimentally derived complex contact angle Θ_{Adv} = 1° + 12.8i° [2], an apparent complex surface tension of water of

## 2 Material and methods

Titanium miniplates either machined [3], [9] or titanium plasma sprayed (TPS) [10] and the acid etching methods for preparation of sandblasted, acid etched (SLA) [3] and chromosulfuric acid etched (CSA) surfaces [9] have been described. Dental implants (SLA, Morphoplant GmbH, Bochum Germany; total length: 13 mm; thread length: 7 mm; ∅ collar: 5 mm; perimeter 13.35 mm [4]) were used. Wilhelmy balance measurements were made on a Tensiometer DCAT 11 EC (Dataphysics, Filderstadt, Germany) with a weight resolution of 10 μg in ultrapure water. The SCAT software package (Vers. 3.2.2.86), which wrongly reports imaginary contact angles as zero instead of undefined, was adapted for calculating baseline corrections [3] and imaginary contact angles [1, 3, 11]. For baseline correction the baseline difference is subtracted from all values of the wetted sample but not from those of the dry sample i.e. advancing baseline [3], [11]. Intrinsic contact angles may be determined on smooth or defined as smooth surfaces [2] dynamically or by a picoliter sessile drop method (OCA 40 Microdevice; Dataphysics) [4]. As of nomenclature [6], classical contact angles in real number space are denoted by a lower case or small theta (*θ*), complex contact angles by an upper case or capital theta (Θ) and imaginary contact angles by a small lambda (*λ*). Contact angles directly calculated from the force measurements without buoyancy correction (*θ*_{V}, Θ_{V}) are called virtual dynamic contact angles [5]. Hybrid dynamic contact angles combine an advancing contact angle in real number space with a receding contact angle in imaginary number space [5]. For other terminology see [5].

### 2.1 Theoretical considerations

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:

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

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:

Where Θ is a complex contact angle, the terms

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

where *F* is the measured net force, *P* is the perimeter of the sample, *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:

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

With the constant

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]:

where *F _{W}* is the Wilhlemy force and

*F*is the sum of all additional forces acting on the miniplate that are not accounted for by the Young equation and

_{Σ}*κ*is a constant.

*F*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

_{Imb}*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].

## 3 Results and discussion

Complex numbers can be displayed in two forms, either in *cartesian form* z = a + bi which for contact angles becomes:

with the intrinsic real part {*θ*_{0}} and the imaginary part {*λ _{i}*} or in

*polar form*as vector diagrams in a Gaussian number plane with the ordinate (Im) for imaginary numbers and the abscissa (re) for real numbers (see Figure 1). The diagonal vector

*r*is the

_{Θ}*modulus*or magnitude of Θ. The vector

*r*${}_{\Theta}$ also forms the

*argument*or phase angle

*φ*(tan

*φ*= Im/Re) with the abscissa and is a non-negative real number —Θ— defined by the following absolute value equation as:

The hydrophilic and hydrophobic ranges can be defined in two ways: (i) *Approach 1*: According to the complex contact angle Θ < 90° = hydrophilic, Θ > 90° = hydrophobic [2] or (ii) *Approach 2*: according to the Young contact angle (eq. 1) *θ*_{Y} < 90° = hydrophilic, *θ*_{Y} > 90° = hydrophobic [6]. Both definitions have their pros and cons but being based on the Young equation appears most reasonable.

Crucial is the directionality of the vector *r* in order to avoid identical imaginary contact angles in the hydrophilic and hydrophobic ranges. In contrast to a previous suggestion [2] the directionality of the modulus can be mathematically based on eqs. 6 & 7 [6]. Calculations show that for

*θ*

_{Y}< 90° the imaginary part is positive and for

*θ*

_{Y}> 90° the imaginary part is negative. In Figure 1A–C the vector of the imaginary part is upward (positive) and in Figure 1C and D the vector of the imaginary part is downward. Thus hydrophilic and hydrophobic complex contact angles are complex conjugates.

As shown below, complex contact angles are of great utility in the analysis of highly wettable microstructured medicinal surfaces e.g. current dental implants.

A dilemma of superhydrophilic surfaces is that they all have the same dynamic contact angle of zero degrees (Table 1) irrespective of the surface roughness (Ra), which varies from 1–2 nm for highly cleaned smooth glass, to 2–3 μm for a SLA surface and up to 30 μm for a titanium plasma sprayed (TPS) surface. It is demonstrated in that according to the novel Wilhelmy evaluation only the quartz glass surface (Table 1A) is ultrahydrophilic (i.e. *θ*_{A}/*θ*_{R} ∼ 0°/0°). Both the SLA- and TPS-surfaces are hyperhydrophilic with the SLA surface (Table 1B) characterized by the dynamic imaginary contact angles Θ_{A}/Θ_{R} = 6.4i° ± 3.2i°/8.3i° ± 2.4i° and the TPS surface (Table 1C) by Θ_{A}/Θ_{R} = 10.9i° ± 2.1i°/13.5i° ± 0.9i°.

Comparison of the classical Wilhelmy (real contact angles) and novel Wilhelmy evaluation (imaginary contact angles) of force measurements on super-/hyperhydrophilic surfaces of varying surface roughness.

Samples of varying roughness | A. Quartz glass surface (Ra ∼ 1–2 nm) [13] | B. Titanium SLA surface (Ra ∼ 2–3 μm, r ∼ 3) [3]_{m} | C . Titanium TPS surface (Ra ∼ 30 μm, r ∼ 20) [10]_{m} | |||
---|---|---|---|---|---|---|

Advancing | Receding | Advancing | Receding | Advancing | Receding | |

Classical Wilhelmy (Eq. 4) | θ_{A} = 0° | θ_{R} = 0° | θ_{A} = undefined | θ_{R} = undefined | θ_{A} = undefined | θ_{R} = undefined |

Novel Wilhelmy (Eq. 8) [10] | θ_{A} = 0° | θ_{R} = 0° | θ_{A} = 6.4i° ± 3.2i° (n = 15) | θ_{R} = 8.3i° ± 2.4i° (n = 15) | θ_{A} = 10.9i° ± 2.1i° (n = 5) | θ_{R} = 13.5i° ± 0.9i° (n = 5) |

Finally an example of a Wilhelmy profile of a dental SLA type implant is shown in Figure 2. Conspicuous is the difference in the baseline level between the advancing and the receding trajectory. This difference of 22 mg is due to water imbibition by the rough surface (eq. 8). Since the water uptake mimics a false force (i.e. contact angle) it has to be subtracted from the weight yielding the imaginary dynamic contact angles Θ_{A}/Θ_{R} = 24.2i°/27.1i° with minimal hysteresis.

**In conclusion:** Although some aspects of complex contact angles are still unclear, they have proven to be very useful in the analysis of highly wettable rough medicinal surfaces.

## Author’s Statement

Research funding: The author state no funding involved. Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent is not applicable. Ethical approval: The conducted research is not related to either human or animal use.

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