A general mathematical form and description of contact angles

Jennissen, H. P.

doi:10.1002/mawe.201400296

Cited by 4 publications

(13 citation statements)

References 35 publications

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“… First the imaginary contact angle (λ) is calculated according to Ref. from the complex contact angle (Θ) and the intrinsic contact angle (θ 0 ) . The complex contact angle pair (θ 0 + λ i) is then converted from [degrees] to [rad]. Taking the secant of the complex contact angle pair (θ 0 + λ i) in [rad] according to Eq. converts the angles to real and imaginary trigonometric numbers

normalP / normalF thinmathspace({normalσ}_{L {normalV}_{normalℂ}} + {σ^{″}}_{L {normalV}_{normalℂ}} i thinmathspace)

. Division of the real trigonometric number by P/F yields the constant

{normalσ}_{L {normalV}_{0}}

= 73 mN/m. Division of the imaginary trigonometric number by P/F yields the desired

{σ^{″}}_{L {normalV}_{normalℂ}}

. …”

Section: Resultsmentioning

confidence: 99%

“…First the imaginary contact angle (λ) is calculated according to Ref. from the complex contact angle (Θ) and the intrinsic contact angle (θ 0 ) .…”

Section: Resultsmentioning

confidence: 99%

“…[11,12] and on methods and nomenclature see Ref. [3,7]. 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) meet at thermodynamic equilibrium according to the equation:…”

Section: Methodsmentioning

confidence: 99%

“…thermodynamic equilibrium on an ideal surface by an ideal liquid. Recently contact angles were described as complex numbers consisting of an imaginary part and a real part, the latter being either the Young or the intrinsic contact angle :

Θ = {{normalθ}_{0}} + {λ i} [°]

where Θ is the observed (effective) complex contact angle, consisting of the real part {θ 0 } (intrinsic contact angle) and the imaginary part {λ} together with the imaginary unit i (imaginary contact angle, ). Taking the cosine of the complex contact angle according to complex trigonometry leads to the equation:

\cos ({normalθ}_{normalY} + λ i) = \cos {normalθ}_{normalY} \cosh λ - i \sin {normalθ}_{normalY} \sinh λ

…”

Section: Theoretical Considerationsmentioning

confidence: 99%

“…After correcting for imbibition imaginary dynamic contact angles have been reported for sandblasted and acid etched (SLA) and plasma sprayed titanium surfaces on miniplates as well as on sandblasted and acid etched surfaces of dental implants . Recently it was shown that imaginary contact angles form a part of complex contact angles . It was proposed that imaginary contact angles result form the non‐ideality of the surface (e.g.…”

Section: Introductionmentioning

confidence: 99%

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On the origin of the imaginary part of complex contact angles

Jennissen

2015

Materialwissenschaft Werkst

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Complex contact angles of water on rough surfaces have been described to consist of a real part for ideal wetting and of an imaginary part for non‐ideal wetting (Jennissen, 2011, 2014). The concept of imaginary contact angles has accordingly been successfully applied to the analysis of hyperhydrophilic rough surfaces. The origin of the imaginary part of complex contact angles has hitherto been attributed to the non‐ideality of surfaces (e.g. surface roughness) but is otherwise unclear. It is generally accepted that the Young equation is valid, if the criteria ideal surface and thermodynamic equilibrium are fulfilled. What has been overlooked is a third criterion, namely, if an ideal fluid also exists. This criterion, has never been seriously questioned, but is assumed to be fulfilled. Recent evidence indicates that this assumption is false. Xiong et al. 2014 have reported that the surface tension of water is a complex quantity consisting of a real and an imaginary part. In this paper it is demonstrated that imaginary contact angles can also originate after incorporation of a complex surface tension of water into the Young and Wilhelmy equations, thus confirming the concept of complex contact angles also from the side of the non‐ideality of the fluid.

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normalP / normalF thinmathspace({normalσ}_{L {normalV}_{normalℂ}} + {σ^{″}}_{L {normalV}_{normalℂ}} i thinmathspace)

. Division of the real trigonometric number by P/F yields the constant

{normalσ}_{L {normalV}_{0}}

= 73 mN/m. Division of the imaginary trigonometric number by P/F yields the desired

{σ^{″}}_{L {normalV}_{normalℂ}}

. …”

Section: Resultsmentioning

confidence: 99%

“…First the imaginary contact angle (λ) is calculated according to Ref. from the complex contact angle (Θ) and the intrinsic contact angle (θ 0 ) .…”

Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Θ = {{normalθ}_{0}} + {λ i} [°]

\cos ({normalθ}_{normalY} + λ i) = \cos {normalθ}_{normalY} \cosh λ - i \sin {normalθ}_{normalY} \sinh λ

…”

Section: Theoretical Considerationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the origin of the imaginary part of complex contact angles

Jennissen

2015

Materialwissenschaft Werkst

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Analysis of the osseointegrative force of a hyperhydrophilic and nanostructured surface refinement for TPS surfaces in a gap healing model with the Göttingen minipig

et al. 2016

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BackgroundA lot of advantages can result in a high wettability as well as a nanostructure at a titanium surface on bone implants. Thus, the aim of this study was to evaluate the osseointegrative potential of a titan plasma-sprayed (TPS) surface refinement by acid-etching with chromosulfuric acid. This results in a hyperhydrophilic surface with a nanostructure and an extreme high wetting rate.MethodsIn total, 72 dumbbell shape titan implants were inserted in the spongy bone of the femora of 18 Göttingen minipigs in a conservative gap model. Thirty-six titan implants were coated with a standard TPS surface and 36 with the hyperhydrophilic chromosulfuric acid (CSA) surface. After a healing period of 4, 8, and 12 weeks, the animals were killed. The chronological healing process was histomorphometrically analyzed.ResultsThe de novo bone formation, represented by the bone area (BA), is increased by approximately 1.5 times after 12 weeks with little additional benefit by use of the CSA surface. The bone-to-implant contact (BIC), which represents osseoconductive forces, shows results with a highly increased osteoid production in the CSA implants beginning at 8 and 12 weeks compared to TPS. This culminates in a 17-fold increase in BIC after a healing period of 12 weeks. After 4 weeks, significantly more osteoid was seen in the gap as de novo formation in the CSA group (p = 0.0062). Osteoid was also found more frequently after 12 weeks at the CSA-treated surface (p = 0.0355). The site of implantation, intertrochanteric or intercondylar, may influence on the de novo bone formation in the gap.ConclusionsThere is a benefit by the CSA surface treatment of the TPS layer for osseointegration over an observation time up to 12 weeks. Significant differences were able to be shown in two direct comparisons between the CSA and the TPS surface for osteoid formation in the gap model. Further trials may reveal the benefit of the CSA treatment of the TPS layer involving mechanical tests if possible.Electronic supplementary materialThe online version of this article (doi:10.1186/s13018-016-0434-6) contains supplementary material, which is available to authorized users.

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Protecting ultra- and hyperhydrophilic implant surfaces in dry state from loss of wettability

Lüers

Laub

Jennissen

2016

Current Directions in Biomedical Engineering

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Ultrahydrophilic titanium miniplates with sandblasted and acid etched (SLA) surfaces were protected from loss of hydrophilicity by an exsiccation layer of salt and stored in a dry state. Various salts in different concentrations were tested in respect to their conservation capacity and optical appearance. Potassium phosphate buffer in a specified composition appeared to be optimal. This optimal system was applied in a long time storage experiment showing no loss of hydrophilicity over years. It was also transferred with success to hyperhydrophilic dental implants.

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A general mathematical form and description of contact angles

Cited by 4 publications

References 35 publications

On the origin of the imaginary part of complex contact angles

On the origin of the imaginary part of complex contact angles

Analysis of the osseointegrative force of a hyperhydrophilic and nanostructured surface refinement for TPS surfaces in a gap healing model with the Göttingen minipig

Protecting ultra- and hyperhydrophilic implant surfaces in dry state from loss of wettability

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