Fluid nanobubbles and adsorption at solid substrates [Elektronische Ressource] = Fluide Nanoblasen und Adsorption auf festen Substraten / Ralf Kaminke. Betreuer: Klaus Mecke

De
Fluid Nanobubbles and Adsorptionat solid SubstratesFluide Nanoblasen und Adsorptionauf festen SubstratenDer naturwissenschaftlichen Fakult¨atder Friedrich-Alexander-Universita¨tErlangen-Nu¨rnbergzur Erlangung des Doktorgradesvorgelegt vonRalf Kaminkeaus WaiblingenAls Dissertation genehmigt von der Naturwissenschaftlichen Fakulta¨t derFriedrich-Alexander-Universit¨at Erlangen-Nu¨rnbergTag der mu¨ndlichen Pru¨fung: 25.07.2011Vorsitzender derPromotionskomission: Prof. Dr. Rainer FinkErstberichterstatter: Prof. Dr. Klaus MeckeZweitberichterstatter: Prof. Dr. Roland RothContentsZusammenfassung 5Abstract 71 Introduction 92 Homogeneous bulk system 132.1 Coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Isobar liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Liquid adsorption at planar substrates 193.1 Homogeneous confined liquid . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Sharp kink approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Modelling the first adsorbed layer . . . . . . . . . . . . . . . . . . . . . . . 303.4 Wetting layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.6 Influence of a variable liquid density . . . . . . . . . . . . . . . .
Publié le : samedi 1 janvier 2011
Lecture(s) : 70
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Source : D-NB.INFO/1015474993/34
Nombre de pages : 145
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Fluid Nanobubbles and Adsorption
at solid Substrates
Fluide Nanoblasen und Adsorption
auf festen Substraten
Der naturwissenschaftlichen Fakult¨at
der Friedrich-Alexander-Universita¨t
Erlangen-Nu¨rnberg
zur Erlangung des Doktorgrades
vorgelegt von
Ralf Kaminke
aus WaiblingenAls Dissertation genehmigt von der Naturwissenschaftlichen Fakulta¨t der
Friedrich-Alexander-Universit¨at Erlangen-Nu¨rnberg
Tag der mu¨ndlichen Pru¨fung: 25.07.2011
Vorsitzender der
Promotionskomission: Prof. Dr. Rainer Fink
Erstberichterstatter: Prof. Dr. Klaus Mecke
Zweitberichterstatter: Prof. Dr. Roland RothContents
Zusammenfassung 5
Abstract 7
1 Introduction 9
2 Homogeneous bulk system 13
2.1 Coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Isobar liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Liquid adsorption at planar substrates 19
3.1 Homogeneous confined liquid . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Sharp kink approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Modelling the first adsorbed layer . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Wetting layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Influence of a variable liquid density . . . . . . . . . . . . . . . . . . . . . . 58
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Surface tension of spherical interfaces 69
4.1 Planar surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Arbitrary curved geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Contact angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Fluid interactions for spherical gas bubbles . . . . . . . . . . . . . . . . . . 74
4.5 Tolman length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Stability of a spherical bubble 79
5.1 Mechanical equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Chemical potential equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 84
6 Stability of nanobubbles on substrates 87
6.1 Spherical cap on a substrate . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Coexisting equilibrium densities . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3 Non-coexisting density inside the cap . . . . . . . . . . . . . . . . . . . . . 98
6.4 Effective Laplace pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.5 Nanoscopic contact angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7 Summary and Outlook 125
3Appendices 127
A Interaction between planar fluid layers 127
B Alternative approaches for the monolayer density 131
B.1 Linear dependence on involved densities . . . . . . . . . . . . . . . . . . . 131
B.2 Langmuir approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
C Alternative approach for the liquid density 135
Danksagung 139
Bibliography 141
4Zusammenfassung
Das Phasenverhalten von Fluiden kann durch angrenzende Substrate auf verschiedene
Weisen beeinflusst werden. Gasmoleku¨le ko¨nnen an einer Wand adsorbieren und einen
flu¨ssigen Film bilden. Da eine Substratoberfla¨che in der Regel rau ist, stellt sie den
Moleku¨len diskrete Adsorptionspl¨atze zur Verfu¨gung. Dies fu¨hrt dazu, dass die Entropie
in der N¨ahe einer Oberfla¨che hauptsa¨chlich durch die Besetzungswahrscheinlichkeit der
Adsorptionspl¨atze beeinflusst wird. Deshalb wird in dieser Arbeit eine Dichtefunktional-
theorie vorgeschlagen, welche die Eigenschaften der Adsorption von Monolagen im De-
tail erfasst, aber auch die Bildung mesoskopischer Filme und Kapillarkondensation unter
hydrophoben Bedingungen behandelt. Sie gibt Aufschluß u¨ber die Abha¨ngigkeit der
Benetzungseigenschaften, der Abzugskr¨afte sowie der Adha¨sion von Temperatur, Luft-
¨feuchtigkeit und Substratst¨arke. Insbesondere ist dies beim Ubergang von Verbindungen,
die aus einzelnen Moleku¨len bestehen, hin zu kondensierten Kapillarbru¨cken mo¨glich.
Fu¨rdieBeschreibung verschiedenster biologischer Systeme ist dies wichtig. Beispielsweise
klebt ein Gecko an einer Wand, weil Adha¨sionskr¨afte zwischen der Wand und dem Fuß
des Geckos wirken. Dies wird durch die komplizierte hierarchische Struktur des Fußes
erreicht.
AufderanderenSeitekanneinSubstratauchzurStabilisierunggasfo¨rmigerNanoblasen
in einer Flu¨ssigkeit beitragen. In einer reinen Flu¨ssigkeit ko¨nnen solch winzige Bl¨aschen
aufgrund des riesigen Laplace Drucks nicht existieren. An der Grenz߬ache zwischen einer
Flu¨ssigkeit und einem Substrat ko¨nnen sich aber solche gasfo¨rmigen Nanoblasen bilden.
Diese nehmen die Form einer Kugelkappe ein. Ein Dichtefunktionalansatz liefert je nach
Gro¨ße der Kugelkappe, Temperatur und Substrateigenschaften ein metastabiles oder sta-
biles Minimum des Großkanonischen Potentials. Damit ko¨nnen experimentelle Beobach-
tungen u¨ber die Existenz von Nanoblasen untermauert werden. Des Weiteren wird ein
effektiver Laplace Druck eingefu¨hrt, der die Substrateigenschaften beru¨cksichtigt. Er
erkl¨art den scheinbaren Widerspruch, zwischen dem in Nanoblasen gemessenen Druck
und einem deutlich h¨oheren Laplace Druck, der aufgrund des geringen Radiuses solcher
Bl¨aschen zu erwarten wa¨re.
56Abstract
Solidsubstratescaninfluencethephasebehaviouroffluidsindifferentways. Gasmolecules
can adsorb on a wall and build a liquid film. Since the surface of a substrate is normally
corrugated, it provides discrete adsorption sites to the molecules. Thus closed to a solid
substrate the entropy ofa fluidis dominated by theoccupation probability ofthisadsorp-
tion sites. So this thesis proposes a density functional theory which captures accurately
the monolayer adsorption properties but covers also the growth of mesoscopic films and
capillary condensation at hydrophilic conditions. This is applicable to calculate the de-
pendence of wetting properties, pull-off forces and adhesion on temperature, humidity
andsubstrate strength, in particular atthe crossover fromsingle particle to capillary con-
densed bridges. This becomes important for the description several biological systems.
For example the gluing of a gecko an a wall is caused by adhesion forces between the wall
and the gecko’s foot, enforced by the sophisticated hierarchical structure of this foot.
On the other hand a substrate can also make the stabilisation of gas nanobubbles in
a liquid solution possible. Due to a large Laplace pressure such bubbles can not exist in
a bulk liquid. But on a liquid solid surface they can form a spherical cap gas nanobub-
ble. Making a density functional approach delivers a metastable or stable minimum of
the grand canonical potential, depending on the size of the cap, the temperature and
the substrate properties. So experimental observations on the existence of nanobubbles
can be confirmed. Further an effective Laplace pressure, which captures the substrate
properties, will be introduced. It solves the discrepancy between the pressure measured
in nanobubbles and a higher pressure expected due to a high Laplace pressure arising
from the small radius of such bubbles.
781 Introduction
Itisdifficulttodescribeasystemconsistingofmanyparticles,becauseofthemanydegrees
of freedom. The most convenient way to deal with them is density functional theory. The
basic idea is to describe the system by a density distribution (~r) instead of considering
each single particle and its interactions with all the other particles. This approach was
originally invented forquantum mechanical electron systems. A first heuristic description
in[Thomas 1927]and[Fermi 1927]reducedthemanyelectronproblemtoasingleelectron
problembyusingtheelectrondensitydistribution˜(~r). Thisdensitydistributionfunction
became the elementary quantity instead of the many body wave function.
The idea of density functional theory was finally invented by [Hohenberg 1964] for
temperatures T = 0K. The Hohenberg-Kohn theorem states, that a unique energy func-
tional E[˜(~r)] of the electron density ˜(~r) exists. This functional includes all interac-
tions between the electrons and also with external fields. It reaches its minimum for the
ground state electron density ˜ (~r). The corresponding energy is the ground state energy0
E[˜ (~r)]. The task in using quantum mechanical density functional theory is finding an0
electrondistributionfunctionthatminimises theenergyfunctional. ThetheoryofHohen-
berg an Kohn was extended to nonzero temperatures by [Merim 1965]. A more detailed
description on the concepts of density functional theory can be found in [Zeller 06].
Density functional theory can also be applied in other fields of physics. It was, for
example, introduced in nuclear physics by [Brack 1985]. However for this thesis its gen-
eralisation for the physics of classical fluids [Evans 1979] is important. The role of the
electron density distribution function ˜(~r) is taken by the density (~r) of the fluid and
the energy functional has to be replaced by a thermodynamic functional. Here, the grand
canonical potential Ω[(~r)] is appropriate. It captures, as the energy functional for elec-
trons,allexternalinfluencesanditisafunctionalofthesingleparticledensity. Minimising
the grand canonical potential functional leads to the equilibrium density distribution for
the thermodynamical system. It can distinguish between a gas and a liquid phase by the
value of the density as the liquid phase causes a higher density than the gas phase. At
low temperatures the liquid and the gas phase can coexist, as both densities lead to the
same value of the grand canonical potential. For increasing temperatures the difference
between the density of the liquid and the gas phase becomes smaller. It finally vanishes
at the critical temperature T . Above T it is not possible to distinguish between a liquidc c
and a gas phase.
In section 2 of this thesis the simplest case of a bulk fluid is considered without any
external influences and without confinement. In that case the functional simplifies to
a function of a density because a homogeneous density can be assumed. These results
will be used for further calculations. So a fluid far away from any other influences like
substratescanbeassumedtobeinabulkstate. Thisis,forexample, thecaseinsection3,
when adsorption ona planarwall is investigated. Experimental observations [Wong 1990]
of adsorption are used to create a model of a sharp kink density profile that can easily be
calculated. This model is not able to predict spatial structures, because they are already
enforcedbythemodel,butitallowstodescribeobservedstructuresquantitatively. Amore
detailed description of the density profile can be obtained by Monte Carlo simulations
[Abraham 1978] and also by theoretical works [Rosenfeld 1989, Roth 02] using density
functional theory, which can capture lateral density variations.
The focus of this thesis is set on the behaviour of the fluid close to the substrate
surface. In the most common density functionals the substrates are usually modelled by
lateralhomogeneousplanewallsandtheadjacentfluidisdescribedbyacontinuousdensity
9bulk fluid ηbulk
Figure 1.1: Close to a corrugatedsubstrate wall
η the entropy of an adjacent fluid is dominated byadsorption layer l l
the occupation probability on discrete adsorption
sites. So in this theses a density functional ap-
monolayer η s proach is presented, that treats the first adsorp-
0
tionlayerwithdiscretesitesseparatelyfromacon-
tinuous liquid adsorption layer and the bulk fluid.
substrate η
s
distribution (~r). However, this disregards that a real substrate has a corrugated surface
whichoffersdiscreteadsorptionsitestothefluidparticles. Soclosetothesubstratesurface
the entropy of the fluid is dominated by the occupation probability of these adsorption
sites. Thus, in the presented density functional the first monolayer of fluid particles is
treated separately, as it is illustrated in Figure 1.1. The following liquid adsorption layer
as well as the bulk fluid remain a homogeneous fluids. So it becomes possible to capture
monolayer adsorption, which happens even on hydrophobic substrates [Beaglehole 1992,
Freund 1999], as well as the formation of mesoscopic liquid films.
In a system where two substrate plates are parallel to each other, the adsorbed liq-
uid film can fill the complete space between the plates if they are close enough. This
phenomenon is called capillary condensation [Evans 1987], where the fluid density in the
whole gap between the plates increases tremendously from a vapour to a liquid state. A
liquid phase appears in that slit although the gas phase would prevail under the same
conditions in a bulk fluid without any confining substrates. This capillary condensation
makestheformationofliquidbridgespossiblewhichcontributetoadhesionforcesandcan
make a gecko glued on a wall [Huber 05], for instance. As adhesion of a gecko’s spatulae
has been observed even on hydrophobic surfaces, it is important to consider the already
mentioned monolayer adsorption in that case.
The phenomenon ofgecko adhesion is caused by the hierarchical structure of a gecko’s
foot, which is illustrated in Figure 1.2. It consists according to [Ruibal 1965, Hiller 1968]
of 400 600m long lamellae. These are made of on average 4:7m wide and 110m
long setae. Each of them can be further subdivided into 100-1000 spatulae, which are
on average 200nm wide and long. For such setae the pull-off force has been measured
for different substrates. The pull-off force is the force that is needed to pull the setae off
from the substrate. It was found that the pull-off force increases with the humidity H.
This gives evidence that liquid bridge formation explains the increase of adhesion with
humidity. Such liquid bridges are caused by capillary condensation.
From a theoretical point of view, the pull-off force can be calculated from the deriva-
tive of the minimised grand canonical potential with respect to the distance between the
considered objects. In this thesis calculations will be made for the pull-off forces between
two walls instead of a wall and a geckos spatulae. Nevertheless this simplified approach
describes the system qualitatively well as it is accounted for monolayer adsorption. A
strong influence onthe pull-offforces is found onthe onset ofcapillary condensation. The
formation of liquid bridges is responsible for a tremendous increase of the pull-off force.
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