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Molecular Dynamics Studies of Nanofluidic Devices

Zambrano Rodriguez, Harvey Alexander

Publication date:2011

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Citation (APA):Zambrano Rodriguez, H. A. (2011). Molecular Dynamics Studies of Nanofluidic Devices. Kgs. Lyngby, Denmark:DTU Mechanical Engineering. DCAMM Special Report, No. S128

Molecular Dynamics Studies of Nanol uidic Devices

Ph

D T

he

sis

Harvey A. Zambrano

DCAMM Special Report no. S128

May 2011

DTU Mechanical Engineering

Section of Fluid Mechanics

Technical University of Denmark

Nils Koppels Allé, Bld. 403

DK- 2800 Kgs. Lyngby

Denmark

Phone (+45) 45 25 43 00

Fax (+45) 45 88 43 25

www.mek.dtu.dk

ISBN: 978-87-90416-55-3

DCAMM

Danish Center for Applied Mathematics and Mechanics

Nils Koppels Allé, Bld. 404

DK-2800 Kgs. Lyngby

Denmark

Phone (+45) 4525 4250

Fax (+45) 4593 1475

www.dcamm.dk

ISSN: 0903-1685

MOLECULAR DYNAMICS STUDIES OF

NANOFLUIDIC DEVICES

A thesis submitted to The Technical University of Denmark (DTU)as partial fulfillment of the requirements for the degree of PhD

(Philosophiae Doctor).Supervisor: Associate Professor Jens Honore Walther

DTU Mechanical Engineering, Technical University of Denmark

Harvey A. Zambrano R.

March 1, 2012

Preface

This dissertation is the result of my research work carried out in the fluidmechanics section, Department of Mechanical Engineering at the TechnicalUniversity of Denmark (DTU) during the period November 2007 to Novem-ber 2010. The project has been made under the supervision of AssociateProfessor Jens H. Walther.

First of all, I am deeply grateful to Jens Walther for outstanding super-vision, great guidance, permanent scientific discussions, mentoring in highperformance computing and nanoscience topics, and continuous encourage-ment during my entire PhD studies. I would also like to thank to PetrosKoumoutsakos at the Chair of Computational science at ETH Zurich for al-lowing me to use FASTTUBE package and for his significant contributionsto my scientific training. Special thanks to Richard Jaffe at NASA AmesResearch Center for his helpful advice on quantum chemistry, his contribu-tions to the development of my research and his kind hospitality during myvisit at California. I wish to thank Ivo Sbalzarini at ETH Zurich, EfthimiosKaxiras at Harvard University and Dimos Poulikakos at ETH Zurich fortheir collaborations in different parts of my research projects. A very specialnotes of thanks are due to my colleagues and friends at DTU: Jakob, Nestor,Johannes, Gabriel, Gireesh, Anas, Wei, Niels, Søren and Signe for useful sci-entific and non-scientific discussions and for making my stay at DTU veryenjoyable. I want to express my thanks to the faculty members of the FluidMechanics section at DTU: Jens Sørensen, Martin Hansen, Wen Shen, KurtHansen, Henrik Bredmose and Knud Erik Meyer for kind discussions on allkinds of different topics specially during lunch breaks. I am grateful to ViviJenssen, Benny Edelsten and specially to Ruth Vestergaard for permanentpractical help. I wish to express my gratitude to Rocio Rivera since thisthesis partially is the result of our shared research dreams during our collegeyears. I would also like to thank to my lifelong friends Carlos, Omar and

i

Miguel for long distance support and encouragement. I am very grateful tomy Parents Juan A. Zambrano and Aida I. Rodriguez for their permanentencouragement, love and extensive support during my entire education, alsothanks to my siblings Habib and Heidi Zambrano for their invaluable mo-tivation and encouragement. Special thanks go to my uncles Pedro, Jaime,Jose and Juan Rodriguez whose ways of life have a significant impact onme. Finally, but not least, I am eternal indebted to my wife Shirley Jo-hanna Habib, without whose love, patience, constant and extensive support,motivation and company this thesis would not have been written. This dis-sertation is dedicated to my grandparents, specially to the memory of: MiguelA. Rodriguez, Juan F. Zambrano and Bertha Mendoza. This research workhas been made possible by funding from the danish research council by theFTP grant No. 274-06-0465 and the computational support from the DanishCenter for Scientific Computing (DCSC).

Technical University of DenmarkDepartment of Mechanical EngineeringMay 24th 2011, Kgs. Lyngby Denmark

Harvey A. Zambrano

ii

Abstract

Nanotechnology and fluid mechanics are two scientific areas where recentprogress has disclosed a variety of new posibilities. The advances in bothfields stablished the grounds for interdisciplinary approaches and recent find-ings promise novel applications that are leading to a technological revolution.Novel nanofabrication techniques have opened up possibilities for the devel-opment of small-scale integrated devices, such as lab-on-a-chip for biochem-ical synthesis and analysis, the integration is achieved by miniaturization ofthe functional elements e.g., of the channels transporting the fluid and ofthe sensors performing the analysis, and as the size of these devices reachesthe sub-micron range we enter the field of nanofluidics. Nanofluidics is de-fined as the study of flows in and around nanosized objects. Modeling oftransport in nanofluidic systems differs from microfluidic systems becausechanges in transport caused by the walls become more dominant and thefluid consists of fewer molecules. Carbon nanotubes are tubular graphitemolecules which can be imagined to function as nanoscale pipes or conduits.Another important material for nanofluidics applications is silica. Nowadays,silica nanochannels are produced in nanometer scale using different nanofab-rication techniques. Silica nanochannels are being implemented in severalnanotechnology applications such as nanosensor devices, nano separators,nanofilters and a plethora of devices for nanobiological and biochemical ap-plications. Experiments at the nanoscale are expensive and time consumingmoreover the time scale associated to several nanoscale phenomena requires avery high time resolution of the devices performing nanoscale measurements.Computational nanofluidics is the enabling technology for fundamental stud-ies, development, and design of such devices. Computational nanofluidicscomplements experimental studies by providing detailed spatial and tem-poral information of the nanosystem. In this thesis, we conduct moleculardynamics simulations to study basic nanoscale devices. We focus our studies

iii

on the understanding of transport mechanism to drive fluids and solids atthe nanoscale. Specifically, we present the results of three different researchprojects. Throughout the first part of this thesis, we include a comprenhen-sive introduction to computational nanofluidics and to molecular simulations,and describe the molecular dynamics methodology. In the second part of thisthesis, we present the results of three different research projects. Fristly, wepresent a computational study of thermophoresis as a suitable mechanismto drive water droplets confined in different types of carbon nanotubes. Weobserve a motion of the water droplet in opposite direction to the imposedthermal gradient also we measure higher velocities as higher thermal gradi-ents are imposed. Secondly, we present an atomistic analysis of a molecularlinear motor fabricated of coaxial carbon nanotubes and powered by ther-mal gradients. The MD simulation results indicate that the motion of thecapsule (inner carbon nanotube) can be controlled by thermophoretic forcesinduced by thermal gradients. The simulations find large terminal veloci-ties of 100 to 400 nmns−1 for imposed thermal gradients in the range of 1to 3 Knm−1. Moreover, the results indicate that the thermophoretic forceis velocity dependent and its magnitude decreases for increasing velocity.Finally, we present an extensive computational study of nanoscale systemsincluding silica substrates and channels, water and air. This study includesthe calibration of a force field to describe the silica-water-air interactions.Moreover, In this study we perform very long simulations of nanoscale sys-tems containing silica, water and air. We investigate the solubility of air atdifferent pressures in silica-water systems. From our simulations we infer alayer with high air density close to silica surface. Furthermore, we conductsimulations to analyze the earlier stage of the capillary filling process of silicananochannels, we focus this study on the roll of air in this system. We findthat air at high pressures can affect the capillarity in silica channels below10 nm height.

iv

Resume

Nanoteknologi og fluid mekanik er to videnskabelige omrader, hvor de senestefremskridt har afsløret en række nye teknologiske og videnskablige mulighed-erne. Fremskridtene i begge felter giver mulighed for tværfaglige metoder ogde seneste resultater lover mulighed for nye anvendelser, der kan føre til enteknologisk revolution. Nye nanofabrikationsteknikker har abnet mulighederfor udvikling af sma integrerede enheder, sa som lab-on-a-chip for biokemisksyntese og analyse. Integration opnas ved miniaturisering af funktionelleelementer f.eks. af kanalerne som tranporterer væskerne og af sensorerne,der udfører analysen. Nar størrelsen af disse anordninger nar sub-micronomrade, taler vi om nanofluidics. Nanofluidics defineres som studiet afstrømninger i og omkring nanoobjekter. Modellering af transport i nanoflu-idic systemer adskiller sig fra mikrofluid systemer, fordi ændringer indenfor transport pa grund af væggene bliver mere dominerende og vædskenbestar af færre molekyler. Kulstof-nanorør er rørformede grafit molekyler,som kan tænkes at fungere som nanoskala rør eller ledninger. Et andetvigtigt materiale til nanofluidics anvendelser er silica. I dag fremstilles silicananokanaler i nanometer skalaen ved at anvende forskellige nanofabrikationteknikker. Silica nanonanokanaler bliver benyttet i flere nanoteknologiskeanvendelse sasom nanosensorer, nano separatorer, nanofilters og et antal an-vendelser indenfor nanobiologi og biokemi. Eksperimenter pa nanoskalaener dyre og tidskrævende og iøvrigt kræver de transiente fænomener, somforbundet med flere nanoskala fænomener en meget høj tidslig opløsningaf maleinstrumenterne. Numeriske beregninger indenfor nanofluidics er enlovende teknologi for grundlæggende studier, udvikling, og design af sadantudstyr. Numeriske beregninger supplerer eksperimentelle undersøgelser oggiver en detaljeret rumlige og tidslig oplysninger om nanosystemet. Denneafhandling omhandler molekylær dynamiske beregning af grundlæggendenanoskala systemer. Arbejdet fokuserer pa undersøgelser af transportmekanis-

v

mer for vædsker og faste stoffer pa nanoskala.Første del af afhandling giver en introduktion til molekylær dynamiske

beregninger. Anden del præsenterers resultater fra tre forskellige forskn-ingsprojekter. Første projekt omhandler et numerisk studie af termoforese,som en egnet mekanisme til at drive vanddraber gennem forskellige typer afkulstof nanorør. Resultaterne demonstrerer at vanddraber kan drives ved atpatrykke et termisk gradient langs nanorørerts akse og at hastigheden voksermed voksende termisk gradient. Det næste projekt omhandler en atomaranalyse af en molekylær lineær motor fremstillet af ko-aksielle kulstofnanorør,som drives ved en termisk gradient. Molekylær dynamiske beregninger viser,at bevægelse af kapslen (det indre kulstof nanorør) kan kontrolleres af ther-mophoretic kræfter fremkaldt af termiske gradienter. Simuleringerne findestore terminale hastigheder pa 100 til 400 nm/ns for palagt termisk gra-dienter i intervallet af 1 til 3 K/nm. Desuden viser resultaterne, at dentermoforetiske kraft er hastighedsafhængige og dens størrelse reduceres vedøget hastighed. I sidste del af afhandlingen præsenteres en omfattende un-dersøgelse af nanoskala systemer bestaende af vand og luft ved silica over-flader, Denne undersøgelse omfatter kalibrering af de molekylær dynamiskekraftfelter til beskrivelse af silica-vand-luft interaktioner. Desuden foretagesmeget lang beregninger af nanoskala systemer, der indeholder silica, vandog luft. Lufts opløseligheden i vand undersøges ved forskellige tryk i silica-vandsystemer. Beregningerne indikerer eksistens af et lag med høj luft kon-centration tæt pa silica overfladen. Endelige foretages beregninger af dentidlige fase af kapillar fyldning af silica nanokanaler. Fokus ved disse bereg-ninger er lufts indflydelse pa fyldningsprocessen, og undersøgelsen viser atluft ved højt tryk kan pavirke kapillaritet i silica kanaler.

vi

Contents

Preface i

Abstract iii

Resume v

1 Introduction 2

1.1 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 4

2 Nanofluidics 6

2.1 Basic nanodevices . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Carbon nanotubes as fluid conduits . . . . . . . . . . . 72.1.2 Silica channels with subnanometer dimensions . . . . . 11

2.2 Driving mechanism for liquids and solids at the nanoscale . . . 132.3 Computational modeling of nanofluidic systems . . . . . . . . 15

3 Molecular Simulations 17

3.1 The molecular dynamics method . . . . . . . . . . . . . . . . 183.1.1 Potential functions . . . . . . . . . . . . . . . . . . . . 203.1.2 Periodic boundary conditions . . . . . . . . . . . . . . 253.1.3 Microcanonical and Canonical ensembles . . . . . . . . 263.1.4 Applications and achievements . . . . . . . . . . . . . . 28

4 Thermophoretic Motion of Water Nanodroplets Confined In-

side Carbon Nanotubes 32

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

vii

5 Carbon Nanotubes Based Molecular Motors Driven by Ther-

mophoresis 52

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Methodology and thermodiffusion study . . . . . . . . . . . . 555.3 Friction and thermophoretic force study . . . . . . . . . . . . 565.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Molecular dynamics study of the interface between silica,

water and air 61

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2 Methodology and force fields . . . . . . . . . . . . . . . . . . . 63

6.2.1 Silica interaction potential . . . . . . . . . . . . . . . . 636.2.2 Water interaction potential . . . . . . . . . . . . . . . . 656.2.3 Silica-water interaction potential . . . . . . . . . . . . 656.2.4 Silica-air interaction potential . . . . . . . . . . . . . . 696.2.5 Water-air interaction potential . . . . . . . . . . . . . . 71

6.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 746.3.1 WCA of water nanodroplets in air at different pressures 746.3.2 High density air layers at the silica-water interface . . . 756.3.3 Study of capillary filling of water in silica nanochannels 78

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Summary and Future Work 88

1

Chapter 1

Introduction

Miniaturization, an ongoing process in many technological fields, is leadingto systems on the nanoscale. As early as 1959, Richard Feynman recognizedthe possibilities and challenges by controlling the motion of matter at thenanoscale in his seminal talk “There is Plenty of Room at the Bottom” [70].Nanotechnology can be defined as the techniques and methods of control-ling matter on an atomic and molecular scale. Nanotechnology deals withstructures with at least one dimensions below 100 nm and a second dimen-sion below 1µm, and involves developing materials or devices within thatscale [46, 18, 142]. Nanodevices are devices with at least one essential func-tional component that is a nanostructure. Nanotechnology utilizes the unitsprovided by nature, which can be assembled and also manipulated based onatomic interactions [197]. Atoms and molecules are therefore the basic build-ing blocks of nanotechnology. Nevertheless, there is a fundamental differencefrom the classical definition of a building material used in a conventional tech-nical environment, which also consists of atoms and molecules in bulk mate-rials. The smallest unit in macroscale terms includes an enormous number ofsimilar atoms and molecules, in contrast to the small assembles of particlesaddressed in nanotechnology. Therefore, the properties of nanostructuresare more closely related to the states of individual molecules, molecules onsurfaces or interfaces than to the properties of the bulk material. A solidsubstrate exposed to the environment is almost invariably covered by a layerof fluid material thus in nanotechnology interfaces between solid, liquid andgas are ubiquitous and of key importance on the behavior of nanodevices.Nanofluidics has been defined as a liquid system in which one has movementand control over fluids in or around nanostructures. Conventional descrip-

2

tion of hydrodynamics of macroscale flows usually are based on the no-slipboundary condition assumption however in microfluidics (flows in channelswith widths of around 100 nm to hundredµm) the no-slip approach has tobe modified. Therefore microflows are typically in the slip flow regime andNavier-Stokes equations with appropriate slip boundary conditions governthese flows. Nevertheless, on the nanoscale, the liquids, gas and substratesexhibit properties not observed in macro and micro fluidics systems thus con-tinuous approaches are often invalid. The different behavior of nanofluidicssystems is due to factors such as the range of molecular interactions, thethermal fluctuations are important and the length scale characterizing hy-drodynamic slip becomes comparable with the system size. Technologically,nanofluidics has become interesting as the basis for further miniaturizationof microfluidic devices down to the nanoscale however nanofluidics is stillan area of intensive research, and many fundamental aspects are subject ofactive debate [205, 182, 89, 122, 173]. Nevertheless, the possibility of ex-tending the lab-on-a-chip concept to the nanoscale, the potential utilizationof carbon nanostructures as a fluid conduits, drug storage and biochemicaldevices, nanofilters and nanosensors are paradigmatic examples of the impor-tance of nanofluidics up to date and of its great potential [46, 92, 180, 222, 90].Nanochannels and nanoscale tubular fluid conduits are the most fundamentalstructures in nano lab-on-a-chip devices. In order to design and manufacturefuture nanoscale lab-on-a-chip devices a full understanding of the flow insidenanochannels and nanofluidic conduits must be addressed including practi-cal mechanisms to drive and control flows at the nanoscale. Today, silicondioxide (silica) and graphite nanostructures have a key place among the mostpromising materials in nanotechnology applications [197, 46, 92, 180, 222, 90].Therefore silica nanochannels and carbon nanotubes must to play a ma-jor role in future development of nanofluidic devices. During the last 50years, computer simulation has become a tool of scientific investigation thatcomplements the traditional methodology of theory and experiments in thediscovery of new scientific and technological knowledge. In fact, computersimulations can be used as an exploratory tool if careful processes of valida-tion are implemented. One of the principal contributions made by computersimulation has been the study of condensed matter, specifically liquids andphase changes [9]. This has been achieved by integrating the motion of anumber of representative particles in the material, this technique is knownas Molecular Dynamics (MD); usually, the motion of the particles is as-sumed to be classical and governed by Newton’s laws although both nuclei

3

and electrons of the atoms are subject to the laws of quantum mechanicssince nuclei are 2000 to 200000 times heavier than electrons then classicalmechanics will provide a much better description for the motion of the nucleithan for the electronic motion. As a detailed description of the electronsis not strictly necessary to describe a molecular system then it is possiblebased on the Born-Oppenheimer approximation [106] to implement a modelwhich the nuclei positions are determinated by changes on the potential en-ergy. Therefore the potential energy is described using force fields whichinclude empiric aproximations of the electronic effects [131, 125, 22]. More-over, quantum effects can be incorporated by including force field parametersand partial charges obtained from quantum chemical calculations using firstprinciples computations (no classical MD) known as ab-initio molecular dy-namics techniques which are too compute-intensive to be useful to simulatelarge systems over interesting time periods. The early work on MD simula-tions was confined to evaluating the properties of hard spheres or using verysimple potential functions however with the continuous increase of the com-putational power, classical MD simulation techniques are becoming suitablefor the study of complex phenomena and capable to handle with simulationsincluding tens of thousands of atoms. Therefore MD simulations could be inthe near future a standard tool to address investigations of nanodevices. Inthis thesis, classical MD simulations are performed to study systems at thenanoscale. The research consists of three major studies. The first study is ananalysis of the transport of water droplets confined inside carbon nanotubesdriven by thermal gradients. The second study is a detailed investigationof a carbon nanotube based thermophoretic linear motor. The third partinvolves the study of the interface between silica, water and air in order tostudy phenomena related to capillary flow of water in silica nanochannels.

1.1 Outline of the Thesis

The remaining part of the thesis is structured as follows:Chapter 2 - Nanofluidics: carbon nanotubes and silica nanochannels as a

fluid conduits. This chapter describes some basic properties and applicationsof the fascinating and challenging novel area of the fluid mechanics at thenanoscale known as nanofluidics. This chapter is focus on the properties,modeling and utilization of structures such as carbon nanotubes and silicananochannels in nanoscale fluidic systems. Furthermore, different types of

4

driving mechanism suitable at the nanoscale to move liquids and solids aredescribed in this chapter.

Chapter 3 - Molecular Simulations. This chapter presents an extensivedescription of the molecular dynamics simulation technique which has beenemployed in the present work to conduct computational studies of nano-fluidic devices. Details about the different MD approaches and their limita-tions, force fields types and calibration processes are discussed. Moreover,In the present chapter some details are included concerning the MD package“FASTTUBE” which has been used in these studies.

Chapter 4 - Thermophoretic Motion of Water Nanodroplets ConfinedInside Carbon Nanotubes. In this chapter MD simulations are performed inorder to investigate the thermophoretic transport of water droplets confinedinside carbon nanotubes. Thermophoresis is the motion of particles drivenby thermal gradients which is also known as Soret effect or thermodiffusion.This investigation includes a systematic study of the motion of water dropletsfor different imposed thermal gradients and for different wetting intensities.Furthermore, the thermophoretic motion is analysed for carbon nanotubesof different chiralities.

Chapter 5 - Carbon Nanotubes Based Molecular Motors Driven by Ther-mophoresis. A molecular linear motor fabricated of coaxial carbon nanotubesand driven by thermophoretic forces is studied using MD simulations. More-over a detailed anlysis of the friction and driving forces is conducted.

Chapter 6 - MD Study of the Interface Between Silica, Water and Air. Inthis chapter interaction force fields are parametrized and calibrated in orderto conduct molecular mynamics simulations of nanofluidic systems involvingamorphous silica, liquid water and air at high pressures. The calibratedMD force field is used to investigate the role of air on the wetting of thehydrophilic silica-water interface including the effect of the presence of airon the capillary filling speed of water in nanochannels.

Chapter 7 - Summary and Future Work.

5

Chapter 2

Nanofluidics

A fluid is a substance that deforms continuously when subjected to a shearstress, irrespectively of the magnitude of the shear stress may be [26, 121]. Al-though fluids consist of molecules, they appear continuous in most macroscalefluid dynamics applications thus the continuum hypothesis which states thatthe properties of a fluid are the same if the fluid were perfectly continuousin structure, can be assumed valid for macroscopic flows. The vast majorityof problems in the dynamics of macroscale fluids are concerned with solvingthe Navier-Stokes equations for incompressible flow assuming the validity ofthe no-slip boundary condition, which states that fluids in contact with asolid surface have the same velocity as the surface so the fluid is not slidingover the solid surface. This condition is called the no-slip boundary condi-tion and represent the most usual way to analyze fluids at the macroscalealthough this condition is an assumption that cannot be derived from basicphysical principles and could, in theory, be violated. For fluids with micro-scale dimensions where the continuum hypothesis is still valid, the no-slipcondition is not suitable [124], therefore, usually a characteristic slip lengthcan be defined. At the nanoscale a contiuum-slip approach is not possiblesince the continuum hypothesis breaks down when the length scale of thesystem under consideration approaches molecular scale which is typical innanofluidic applications. Since a fluids are composed of molecules, one hasthe option of calculating its static or dynamic properties by computing themotion of their constituents [154, 23, 73, 9, 125]. For most purposes sucha procedure is very inefficient, because it provides detailed information atmolecular length scales, which are far beyond the usual area of interest forcontinuum approaches in fluid mechanics. There are, however, situations

6

where the molecular details of a fluid flow (e.g., nanofluidics) are interestingif not crucial. For example, fluids in nanoscopic geometries may exhibit de-viations from the continuum equations, and one may wish to calculate sucheffects from first principles. The usual way to address such issues is throughexperiments or statistical mechanical calculations, but these methods havetheir own limitations [161], and molecular simulations can provide alternativeinsights and results. Moreover, the behavior of thin liquid films, in particularfilms spreading on solid surfaces in presence of air, can involve these smallscales and, additionally, can involve details of the solid-liquid interactionwhich are hidden in bulk fluid mechanics approaches. These cases exemplifyproblems in which a nanoscopic calculations can provide new insights as wellas hard-to-obtain quantitative information to complement other techniques.

2.1 Basic nanodevices

2.1.1 Carbon nanotubes as fluid conduits

Carbon nanotubes are tubular carbon molecules which can be imagined tofunction as nanometer size pipes. In this respect, they could mimic trans-membrane pores and channels and could be used to manipulate the transportacross the membranes of cells, moreover they could be used to filter, storageand control flows in nanometer scale nanofluidic devices [133, 159, 74, 112,210, 136]. Although, its discovery remains a contentious issue. Carbon Nano-tubes (Figure 2.1 and Figure 2.2) have been subjects of intensive researchsince their “discovery” in 1991 by Iijima at NEC laboratories [101]. Carbonnanotubes represent a form of sp2-bonded carbon, in addition to the well-known graphite and the fullerene molecules as illustrated in Figure 2.1. SinceSWCNTs can be considered as a rolled up plane of graphite (Figure 2.3), alsocalled graphene [30, 97, 77], one has to start with the structure of graphene.Figure 2.4a shows the hexagonal structure of a graphene sheet. A roll-upvector can be defined in this sheet by simply taking two points of the lattice.Rolling the sheet up and bringing those two points together creates a single-wall carbon nanotube (SWCNT). The diameter of SWCNTs usually rangesbetween 0.4 and 3 nm. The roll-up vector C can be expressed in terms ofmultiples n and m of the two Bravais lattice vectors of the graphene sheeta1 and a2 as shown in the following equation: C = na1 + ma2, where theintegers n and m define the chirality of the nanotube. A 30◦ angular section

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Figure 2.1: Graphene nanostructures which can be implemented innanodevices, figures from Smalley Group web page at Rice University(http://cohesion.rice.edu/naturalsciences/smalley).

8

Figure 2.2: Multi-wall carbon nanotube observed by Transmission ElectronMicroscopy using a 300 kV microscope (TITAN ANALYTICAL) at the Cen-ter of Electron Nanoscopy at DTU.

9

(a) (b)

Figure 2.3: Graphene nanostructures. (a) Graphite molecular structure. (b)Rolling up process of a graphene sheet.

is sufficient to describe all SWCNTs because of the hexagonal symmetry ofthe honeycomb lattice and the chiral symmetry of (n,m) and (m,n) tubes(Figure 2.4). This 30◦ section of the hexagonal lattice is spread betweentwo particular nanotube configuration, which are symmetric with respect tothe tube axis: the zig-zag nanotubes with m = 0; and the armchair nano-tubes with n = m. All other tubes with arbitrary values of n and m areasymmetric and are, therefore, called chiral nanotubes. Figure 2.4a showsan overview of the different types of CNTs. Depending on their chirality,nanotubes exhibit different behaviors [56]. Generally, a (n,m) nanotube will

have a band gap unless |n−m|3

yields an integer value. According to thisrule the armchair (n, n) nanotubes should always be metallic, but zig-zag(n, 0) and chiral nanotubes can be either metallic or semiconducting [56].In industrial condition, those three types are produced simultaneously andcan not be easily separated. It is difficult to determine chirality of carbonnanotubes produced specially for multi-wall carbon nanotubes. That is thereason why only metallic and semiconducting carbon nanotubes are usuallydistinguished. CNTs have extremely high mechanical strength, which, whencombined with their ability to provide a conduit for fluid transport at molec-ular length scales, makes them attractive candidates for implementation infuture micro- or nanofluidic devices [133, 159]. Therefore, understandingfluid behavior in CNTs is important for the proper design and efficient oper-ation of such systems. Experimental studies of the water-carbon interface innanopipes of CNTs have been performed [79, 134, 159] using TransmissionElectron Microscopy (TEM). Good wettability of the inner carbon walls was

10

(a) (b)

Figure 2.4: Chirality of Carbon nanotubes. (a) Orientation of the roll-upaxis for different types of CNTs. (b) Orientation of carbon atoms on thegraphene sheet edge for different types of CNTs.

shown [79, 134, 159] as illustrated in Figure 2.5. Thus, currently, a very ac-tive topic is the researching of driving mechanism to move and control liquidsinside CNTs [17, 208, 52, 94, 53, 165]. In chapter 4 and 5, thermodiffusionis studied using molecular dynamics simulations as a suitable mechanism tomove flows and solid capsules inside CNTs [215, 214].

2.1.2 Silica channels with subnanometer dimensions

Nowadays, advances in microfabrication and miniaturized analysis have re-sulted in increasingly sophisticated fluidic systems that are fulfilling thepromise of true micro labs-on-a-chip systems by integrating of different pro-cessing steps on a single device. Miniaturization, an permanent in advanceprocess, will finally lead the lab-on-a-chip concept to structures on the nano-scale [1]. Incorporation of biological and synthetic nanochannels in fluidicdevices holds great promise for new analytical applications because there areforces and phenomena at are absent or negligible in larger devices. Some ofthe earliest work in nanofluidics used protein pores in lipid bilayers to char-acterize a range of molecules, including polymers, nuclei acids, metal ions,and organic molecules. Protein pores provide excellent reproducibility withrespect to their dimensions and internal chemistry. However, lipid bilayerson-chip are relatively fragile compared to the silica and polymer substrates

11

Figure 2.5: TEM images of water confined inside carbon nanotubes [159].

12

typical for microfluidic applications. As a result, many research teams arepursuing novel nanofabrication techniques to produce channels in hard andsoft materials with nanoscale size [135, 147]. We are surrounded by silicon ox-ides from sand to the surface of silicon based electronic devices. Silica existsin a very large number of different forms. Each type of silica is character-ized by a structure (cristalline or amourphous) and specific physical chemicalsurface properties which leads to a noticeable set of applications includingseveral nanotechnology devices. Silica can be hydrophobic, as occurs whenthe surface chemical groups are mainly siloxane, or hydrophilic, when the sur-faces expose silanol groups. Hydrophobic silica can be rendered hydrophilicby hydroxylating the siloxane groups into silanol groups. This reaction isreversible thus hydrophilic silica can be transformed into hydrophobic silicaby dehydroxylation heating the surface above 300K [108]. Owing to thewide range of novel applications, amorphous silica has an exceptional placeamong the most promising materials in nanotechnology. The utilization ofsilica nanochannels in future nanofluidic devices depends on the completeunderstanding of the liquid and gas flow throughout this kind of conduits.Many effort has been devoted to study the interaction between silica andwater [123, 57, 58, 195, 152, 60, 87, 69] and the flow throughout silica poresand channels [145, 85, 140, 139, 80, 184, 222], however many fundamentalquestions are, up to date, subject of intense and open debate. Silica Nano-channels with heights below 100 nm are expected to be present in many novelnanofluidics devices. Recently, in a number of different experimental investi-gations, systematic deviations from classical predictions have been measuredfor channels with subnanometer size [145, 147, 184, 182]. In this thesis weattempt to study the capillary filling of subnanometer channels of amorphoussilica from a computational simulation approach.

2.2 Driving mechanism for liquids and solids

at the nanoscale

Today, nanoscale carbon pipes [133, 208, 79, 74, 127] and silica nanochannels [84,145, 182, 184] can be readily fabricated using different techniques [147, 133,84, 56] and they have useful applications related to the transport, delivery andstorage of fluids and gases at the nanoscale. Thus, a complete understandingand controlling of the mass transport through channels with widths of a few

13

nanometers could provide a new set of tools and basic functional elements tobe implemented in future nanodevices. Liquids and solids in nanochannelsmay be driven by electrophoresis, electro-osmosis, osmosis, gradients in thesurface tension or Marangoni effect, pressure gradients, and thermophoresis.Electrophoresis [2, 71] is the motion of particles driven by electric forces pro-duced by the presence of an electric field. Electrophoresis has been usedin several nanotechnology applications such as mass nanoconveyors [156],chemical detectors [114], chemical separation devices [216], and transport ofelectrically charged particles throughout carbon nanotubes [190] and nano-channels [175]. Electro-osmosis [168, 36] is the motion of polar liquids inducedby an electric potential imposed across a porous, capillary tube, channel ormicropipe. Electro-osmosis is a prominent driving mechanism in micro andnano flow conduits therefore some nanofluidic applications have been de-veloped [45, 136]. Osmosis is the movement of water molecules across aselectively permeable membrane from an area of low solute concentration toa zone high solute concentration. When a solute gradient is applied acrossan impermeable porous membrane the osmotic flow rate will be equal to thatcreated by an equivalent pressure gradient, and that it will be independentof the shape or the size of the solute [50]. Osmosis is important in biologicalsystems, as many biological membranes are semipermeable. Kalra et al. [112]have investigated the water flow driven by a osmotic gradient between twosemipermeable carbon nanotubes membranes that separate compartments ofpure water and concentrated aqueous salt-solution. Marangoni effect or gra-dients in the surface tension [169] have been used to transport liquid acrossarrays of aligned carbon nanotubes [221]. Molecular dynamics simulationshave been performed to investigate the water flow through carbon nano-tubes driven by pressure gradient [185, 186]. In chapters 4 and 5 of thisthesis we study the thermophoretic motion of liquids [215] and solids [214]confined inside carbon nanotubes. Thermophoresis or thermal diffusion isthe motion of mass driven by thermophoretic forces induced by thermal gra-dients [78, 149, 150, 62, 165]. Experimental studies have demonstrated thatthermophoresis is a suitable mechanism to move and control mass at thenanoscale [17, 183]. Moreover, molecular dynamics studies have shown thatthermal diffusion can indeed become a prominent, accurate and fast mecha-nism to drive mass throughout carbon nanotubes [160, 215, 214, 165, 166].

14

2.3 Computational modeling of nanofluidic sys-

tems

Experimental studies of nanofluidic systems are difficult to implement, usu-ally include expensive instrumentation and very often detailed measurementare not affordable due to the length and time scales involved in nanofluidicphenomena. Computational simulations have been performed to providefundamental understandings of nanoscale phenomena showing its potentialusability to the wide range of nanofluidic applications [116, 3, 39, 49, 51,112]. Classical Navier-Stokes solutions modeling macroscopic flows havebeen widely used in numerous fluid engineering fields. However, a fun-damental difficulty with continuum models is that they can not representthe atomistic ordering of the fluid structure near to a solid surface there-fore we cannot expect the Navier-Stokes equations to accurately describeflow through very narrow channels or pores. Thus for nanoconfined fluids,molecular simulations have shown that the Navier-Stokes theory shows asignificant deviation from atomistic approaches and its hypothesis are notsatisfied [141, 189, 188, 191, 211, 184]. During the past decade moleculardynamics simulations have emerged as a powerful tool for probing the mi-croscopic behaviour of liquids at interfaces [117]. With the advance of super-computing resource, molecular dynamics simulations have became one of themost promising methods to provide a clear and fundamental understandingof fluidic devices at the molecular level. We have seen that molecular sim-ulations, which describe tiny pieces of matter over very short time intervalsprovide a description of fluid flows over a range of length scales from sub-continuum up to the point of quantitative agreement with the Navier-Stokesequations [117, 199]. The surprising aspect of this statement is that extremelysmall regions of fluid show bulk behavior. The key point of the methodologyis that it simultaneously provides information missing from the continuumapproaches. Molecular dynamics simulations are analogous to a very pre-cise virtual microscopic laboratory where complete resolution is available. Inthis thesis, molecular dynamics simulations are performed in order to studynanofluidics systems. A detailed description of the molecular dynamics tech-nique is presented in chapter 3. The set up of the simulations, details of thepotentials and parameters implemented are presented in the correspondingchapters (chapter 4, chapter 5 and chapter 6). In general, we attempt toimplement models and techniques to reduce the computational cost due to

15

phenomena such as the spreading of water droplets on hydrophilic surfaces,the annealing of silica slabs, the dissolution of air in water and the watercapillary filling of silica nanochannels in presence of air involve time scalewhich are close the current limit of the available computational resources.Nevertheless, we have to implement models that describing with sufficientdetail the complexity of the phenomena involved in order to reproduce themain properties that we attempt to measure. In this thesis, we implementand develop computational models and conduct large scale molecular dy-namics simulations of basic nanosystems attempting to contribute to theunderstanding of the capillary filling process of silica nanochannels and tothe exploitation of thermophoresis to drive and control solid and liquids atthe molecular length scale. We perform classical molecular dynamics simula-tion using the MD package FASTTUBE, which is described in the chapter 3and a more detailed description of the development of the code can be foundin Walther et al. [198] and Werder et al. [206, 207].

16

Chapter 3

Molecular Simulations

In the last four decades computational molecular simulations [6, 7, 5, 9, 73,23] have made immense progress and nowadays are becoming an essentialtool in many areas of science and technology. It is due to the continuousincreasing in the power of digital computers and, likely to a larger extent,to the development of new methods and models for simulation of complexfluids and materials. Computer simulations act as a bridge between molecu-lar length and time scales and the macroscopic world, computer simulationsrequire a representation of the interactions between molecules, and providepredictions of bulk properties. The predictions are “exact” in the sense thatthey can be made as accurate as we like, just subject to the limitations im-posed by our computational capabilities. Simultaneously, information aboutthe hidden detail behind bulk measurements can be obtained. Moreover sim-ulations act as a link between theory and experiments. Indeed, a theory canbe examined by performing a simulation using a tested model. The modelcan be previously validated by comparing with experimental results. Fur-thermore, simulations can achieve details of the properties of a system underconditions that are difficult or impossible to reproduce in the laboratory e.g.,working at extremes of temperature or pressure. In general, a molecularsimulation consists of a computational representation of a system in whichrealistic molecular positions are used to extract structural, thermodynamicand dynamic information. The term “position” denotes a set of Cartesiancoordinates for all the atoms or molecules that constitute a system. Thereare two main families of molecular simulation techniques [73, 9], which aremolecular dynamics [6, 7, 5] and Monte Carlo [4, 73, 9]. Additionally, thereexist a group of hybrid techniques which combine features from both [40, 23].

17

In this thesis, only molecular dynamics simulations have been performed thusin the present chapter, we shall concentrate only on this kind of simulations.For simulations of fluids at the nanoscale the obvious advantage of molec-ular dynamics over Monte Carlo simulations is that it gives a route to dy-namical properties of the system, e.g., transport coefficients, time-dependentresponses to external properties variations, rheological properties and spec-tra [73, 170]. In molecular dynamics simulations the positions are obtainedby numerically solving of differential equations of motion and, hence, thepositions are connected in time and provide information of the dynamics ofindividual molecules as in a motion picture [9]. MD simulations compute themotions of individual molecules. They describe how positions, velocities andorientations change in time. The fundamental idea is that the behaviour ofa system can be obtained if we have a set of initial positions and forces ofinteractions between the particles of the system. The corresponding motionequations can only be solved numerically due to the classical N -body problemlacks a general analytical solution thus molecular dynamics is a methodologywhose appearence is directly linked to the origins of the digital computerage. However, the theoretical basis for molecular dynamics rests on many ofthe results produced by the great names of mechanics such as Euler, Hamil-ton, Lagrange and Newton. In a molecular dynamics simulation, one oftenwants to measure the macroscopic properties of a system through micro-scopic simulations thus the connection between microscopic simulations andmacroscopic properties is made via statistical mechanics which provides theneeded mathematical expressions that relate macroscopic properties to thedistribution and motion of the atoms and molecules of the N -body system.

3.1 The molecular dynamics method

Molecular dynamics consists basically of the simultaneous motion of a num-ber of atomic nuclei and electrons forming a molecular entity. Strictly speak-ing, a complete description of such a system requires solving the full time-dependent Schrodinger equation including both electronic and nuclear de-grees of freedom. This, however, is a formidable computational task whichis in fact unfeasible, at present, for systems consisting of more than threeatoms and more than one electronic state [23]. Therefore, the concept of themolecular dynamics method rests on a very important simplification in quan-tum mechanics, the Born-Oppenheimer approximation [23, 125, 106] which

18

is a simplification of the behaviour of a system of nuclei and electrons. Thenuclei mass is much larger than the mass of the electrons and its velocityis correspondingly small. Thus, the nucleus interacts with the electrons asif they were a charge cloud, while the electrons feel as if the nuclei werestatic. Thus, the electrons are moved “instantly” to any position of thenuclei. The molecular dynamics approach rests on the Born-Oppenheimerapproximation [23]. There exist different types of molecular dynamics sim-ulations which could be divided in two main families first-principles or abinitio molecular dynamics [37, 106] and classical molecular dynamics. In abinitio molecular dynamics method the electronic behavior can be obtainedfrom first principles by using a quantum mechanical methodology [37], suchas Density Functional Theory (DTF) [23, 81]. Although first-principles cal-culations provide important information that is not available from classicalmolecular dynamics methods [81], such as density of electronic states andchemical reactions, due to the cost of treating the electronic degrees of free-dom, the computational cost of this simulations is much higher than classicalmolecular dynamics [23, 125]. Consequently, ab initio molecular dynamicssimulations are limited to smaller systems and shorter periods of time [23].The basis of this work is the technique of classic molecular dynamics com-puter simulation. The classic molecular dynamics technique [6, 7, 9, 73]consists of the numerical solution of the classical equations of motion, whichfor a simple atomic system with N elements may be written,

−→fi = mi

−→ai (3.1)

−→fi = − ∂

∂−→riµ(ri, rj) (3.2)

where−→fi denotes the force acting on the i-th atom due to the interactions

with the other atoms, mi is the mass of the atom, −→ai is the acceleration

produced in the atom by the force−→fi and µ is the potential energy which

describes the interaction between the atoms. Therefore, we need to be able

to compute the forces−→fi , acting between the atoms which can be derived

from a potential energy function µ(ri), using the derived force we can obtainthe accelerations and finally, integrating the accelerations, we are able tocalculate the new velocities and positions of the atoms. In our simulationswe use the molecular dynamics package FASTTUBE which discretize theequations of motion using the leap-frog time integration scheme [93].

19

3.1.1 Potential functions

The calculation of the force acting on every particle is the most time consum-ing part of a molecular dynamics simulation [73, 161]. These forces shouldbe derivated from a potential energy function.

µ(rN) = µbonded + µnon−bonded (3.3)

The part of this potential function µnon−bonded representing non-bonding in-teractions between atoms is usually split into one-body, two-body and three-body terms:

µnon−bonded =

N∑

i

ν(ri) +

N∑

i

N∑

j>1

µ(ri, rj) + ... (3.4)

It should be noted that ri is the separation between adjacent pairs of atoms.The ν(r) term represents an externally applied potential or the walls of acontainer, it is usually dropped by imposing periodic boundary conditions inthe simulation of a system [9, 23]. Therefore, it is usual to focus on the pairpotential µ(ri, rj) = µ(rij) and neglect (as the nature of system allows thiskind of simplification) three body (and higher order) interactions [9, 125].The interactions, at the simplest level, occur between pair of atoms andare responsible for providing the two principal features of an inter-atomicforce. The first is a resistance to compression, hence the interaction repelsat close distance, and the second is to bind the atoms together [125, 131].Potential functions exhibiting these characteristics can adopt a variety offorms and, when chosen carefully, actually provide useful models for realsubstances [9, 125]. There are different ways to obtain these potential func-tions or force fields. These quantities can be derived from calculations basedon first principles (ab initio calculations or density functional theory) by cal-ibrating the potential parameters to the values obtained from solving theelectronic structure therefore obtaining new parameters for the classical po-tentials [87, 81]. Another possibility is to fit the potential parameters to val-ues obtained from experimental measurements [206, 49]. In fact, in chapter 6,we follow this methodology to parametrize a potential model for silica-water-air as discussed later. In this section some types of potential energy functionare described. A MD simulation requires appropriate functions describingthe potential energy between the atoms and molecules present in the system.The intermolecular interactions can be separated in two types, bonding and

20

non bonding. Non bonding interactions include two types of contributions,the long range (Coulomb interactions) and the short range forces (Van derWaals interactions).

Van der Waals interactions

Van der Waals forces can be interpreted as the part of the non bondinginteractions which are not related to the electrostatic energy due to atomicelectric charges. Van der Waals forces consist of electrostatic interactionsbetween permanent charge distributions and induced charge distributions,and the ever present forces due to the oscillations of the orbital electrons andthe interaction with the induced dipoles in the neighboring molecules.

Lennard-Jones interaction potential The most simple to implementand one of the best known potential functions for non-bonding interactionswhich was originally proposed for the study of the properties of liquid ar-gon, is the Lennard-Jones (LJ) potential [110, 9]. A pair of neutral atoms ormolecules are subject to two different forces on the border of a large separa-tion and small separation, an attractive force acts over long distances (VanDer Waals force or dispersion force) and a repulsive force acting small dis-tances (the Pauli repulsion or the Pauli exclusion principle) as illustrated inFigure 3.1. Therefore, for a pair of atoms a and b located at ri and rj, thepotential energy is:

Uab(rij) = 4εab

[

(

σab

rij

)12

−(

σab

rij

)6]

, (3.5)

where rij is the distance between the center of the interacting atoms, εabgoverns the strength of the interaction and σab defines a length scale relatedto the equilibrium separation of a pair of atoms; the interaction repels at

short range, then attracts. Due to the relatively fast decay of the(

1

r6ij

)

term

it is custom to truncate the Lennard-Jones potential at some finite cutoffdistance, which usually is fixed at rc = 2.5 σab. There are not theoreticalarguments for choosing the exponent in the repulsive term to be 12, this is acomputational convenience due to it is not possible to derive theoretically afunctional form of the repulsive interaction. Often the Lennard-Jones poten-tial gives a reasonable approximation of a realistic intermolecular potential.However, from exact quantum ab initio calculations has been inferred that

21

Figure 3.1: Interaction energy of molecular argon. Computations usingLennard-Jones potential and experimental resutults. The graphics is fromR. A. Aziz, J. Chem. Phys., vol. 99, 4518 (1993)

22

an exponential type repulsive potential is often more appropriate. There istherefore some justification for choosing the repulsive term as an exponentialfunction, also known as a Buckingham or Hill type potential.

Coulomb potential model

During the decade of the 1780s, Charles Coulomb carried out a set of remark-able experiments [47]. Using these experimental results he showed that theforce between electrically charged bodies separated a distance large comparedto their dimensions varies directly as the magnitude of each charge, variesinversely as the square of the distance between the charges, is directed alongthe line between the center of mass of the charged bodies, and is attractiveif the bodies are charged with opposite charges and repulsive if the bodiesare charged with the same type of charge. All partial charges present in amolecular system interact through a Coulomb potential UC

UC(rij) =qaqb

4πε0rij, (3.6)

where qa and qb represent the partial charges on atoms of type a and b re-spectively, ε0 the vacuum permittivity, rij the inter-atomic distance. SinceCoulomb interactions decay slowly in comparison to van der Waals interac-tions, Coulomb force computation is the most time consuming part of theforce computing process in molecular dynamics [161]. A variety of methodshave been developed to handle with the problem of the long range interac-tions in molecular dynamics. The most simple method is the implmentationof a cut off distance which means that the forces are computing until a fixeddistance afterward the interaction is neglected. The cut off method is verypopular in MD simulations however is not suitable in many systems andphenomena where long range electrostatic forces are important such us ionicliquids, electroosmotic flow and electrophoresis [10, 111, 115]. Methodologiessuch us Ewald summation, cell multipole method and particle mesh approachhave been studied in considerable detail by Frenkel and Smit [73] and Allenand Tildesley [9]

Born-Huggins-Mayer interaction potential The Buckingham poten-tial describes the repulsion force that originates from the Pauli exclusionprinciple by a more realistic exponential function of distance, in contrast tothe inverse twelfth power used by the Lennard-Jones potential [106, 125].

23

Born-Huggins-Mayer potential function is a variation of the Buckingham po-tential which includes a Coulomb term added to a Buckingham term, thistype of function has been implemented successfully to perform moleculardynamics simulations of silica [192, 82]

µab(rij) =qaqb

4πε0r2ij

+ αab exp

(−rijρab

)

− Cab

r6ij, (3.7)

where qa and qb represent the partial charges on atoms of type a and b respec-tively, ε0 the vacuum permittivity, rij the inter-atomic distance and, αab, ρaband Cab represent the Buckingham force field parameters. The Buckinghampotential has a problem for short interatomic distances where it turns over.As rij goes toward zero, the exponential term becomes a constant while thenegative term goes to infinity. Minimizing the energy of a system that has avery short distance between two atoms will thus result in an artifact process(nuclear fusion). Therefore, precautions have to be taken for avoiding thissituation when using Buckingham potential functions.

Bonding interaction potential

For molecules it must consider the intramolecular bonding interactions. Thesimplest molecular model includes terms of the following kind:

µintramolecular =1

2

∑

bonds

krij(rij − req)

2 +

1

2

∑

angles

kΘ

ijk(Θijk −Θeq)2 +

1

2

∑

torsion

∑

m

kΦ,mijkl (1 + cos(mΦijkl − γm)) (3.8)

The geometry is illustrated in Figure 3.2. The bonds involve the separationrij = ‖ri − rj‖ between adjacent pairs of atoms in a molecular framework.In the molecular dynamics studies described in chapters 4 and 5 the carbonnanotube lattice is modeled using a Morse bond, harmonic cosine of thebending angle [125], and a two fold torsion potential developed by Waltheret al. [198], details of the potential function and parameters can be found inchapter 4 of this thesis and Walther et al. [198] and Zambrano et al. [215].

24

Figure 3.2: Geometry of a simple molecule [9].

3.1.2 Periodic boundary conditions

Imposing periodic boundary conditions the simulated system is exactly repli-cated in three dimensions providing a periodic lattice [154]. In general, we areconcerned with molecular systems with a size much larger than the system wecan afford to simulate. The implementation of periodic boundary conditionsreduces the computational cost of a simulation because computations involv-ing intermolecular separations greater than the length of the computationalbox are not included [9]. Moreover small sample size means that, unless sur-face effects are the particular interest, periodic boundary conditions need tobe used. Consider 1000 atoms arranged in a 10 × 10 × 10 cube. Numerousatoms are on the outer cube faces thus these atoms will have a significanteffect on the measured properties. Even for a larger system i.e., including106 atoms, the surface atoms amount to 6% is still not trivial. Surroundingthe cube with replicas of itself takes care of this problem [154]. Whether therange of the potentials is not too long, it can be adopted the minimum imageconvention which consider that each atom interacts with the nearest atom orimage in the periodic system. Nevertheless, attention can be must be paid

25

to the case where the potential range is not short and when it is importantto consider properties of the interfaces or free surfaces. Periodic boundaryconditions are imposed to the systems simulated in our molecular dynamicsstudies in chapters 4, 5 and 6. The implementation of periodic boundaryconditions prevent the unwanted influence of an artificial boundary like thepresence of vacuum or a solid wall however attention should be paid to thissituation due to periodic boundary conditions also can add some artifactsrelated to the periodicity itself [9]. Only if one wish to analyze a crystal,periodic boundary conditions are natural because they suppress all motionsin the rest of the cells that are different from the motions in the originalcomputational box. Indeed, periodic boundary conditions replicate all ofphysical conditions imposed in the original cell thus if one wish to simulate asystem which is under the influence of a no periodic physical property thenspecial care should be taken e.g., as a localized thermal gradient is imposedto a periodic system (Zambrano et al. [215]) the same gradient will be repli-cated in all of replicated cells thus potentially introducing an artificial hightemperatures gradient at each cell boundary.

3.1.3 Microcanonical and Canonical ensembles

An ensemble in statistical mechanics is a theoretical tool which is used foranalyzing the evolution of a thermodynamic system. The ensemble consistsof a collection of copies of the system of interest with a set of fixed andknown thermodynamic variables. For example, the microcanonical ensembleis a thermodynamically isolated system, where the fixed variables are thenumber of particles in the system (N), the volume of the system (V), and theenergy of the system (E). Therefore, the microcanonical ensemble consists ofa set of M-systems each characterized by fixed N, V, and E. The subsystemwithin the ensemble may be in a different microscopic state (microsystem).Nevertheless, each microsystem shares the same fixed thermodynamic prop-erties, for the microcanonical one N, V, and E. A different type of statisticensemble is needed in order to analyze thermodynamically non-isolated sys-tems. The canonical ensemble (NVT) is an idealization of a system whichis allowed to interchange energy with a thermal source at constant tempera-ture while keeping fixed the number of particles (N), the volume (V) and thetemperature of the system (T) therefore the system in a canonical ensembleis in thermal equilibrium state with a heat source. In MD simulations, tra-ditionally, the Energy is conserved when Newton’s equations of motion are

26

integrated. If the volume (V) and the number of particles N are constant inthe simulation, then the simulation represents the microcanonical ensemble(NVE). MD simulations can be performed sampling different ensembles thanthe NVE such as the canonical ensemble where a thermostat or heat bathis coupled to the system in order to obtain a constant temperature (NVT).The temperature in MD simulations is computing using the principles of sta-tistical mechanics. For a system including N atoms and subject to molecularconstrains, the temperature can be obtained from the average of the kineticenergy following the equation:

T =2Ek

(3N −Nc)kB(3.9)

where Ek is the time average of the kinetic energy, N is the number of atomsin the system, Nc is the total number of the internal molecular constrains andkB is the Boltzmann constant. The most widely used thermostats includethe Andersen [11] and the Nose-Hoover [96] thermostats. An alternative wayto keep the temperature fixed is to couple the system to an external heatbath [25] which is maintained at the target temperature. Heat is supplied orremoved as appropriate by the thermostat, which acts as a source of thermalenergy. Velocities of particles in the system are modified at each time step,such that the rate of temperature variation is proportional to the differencein temperature between the thermal energy source and the system.

dT (t)

dt=

[Tsource − T (t)]

τ, (3.10)

where T(t) is the temperature at time t, Tsource is the temperature at theheat bath and τ is a coupling parameter whose value determines the strengthof the coupling between the system and the thermal source. The resultingscaling of the temperature at each time step elapsed is:

∆T = (δt

τ)[Tsource − T (t)], (3.11)

where δt is the integrating time step. this method gives an exponential decayof the temperature of the system towards the target temperature. Therefore,the scaling factor for the temperatures is:

λ2 = 1 +δt

τ[(Tsource

T (t))− 1]. (3.12)

27

If τ is large, then the coupling will be weak. A great advantage of thisapproach is the possibility to equilibrate molecular systems by conducting asystematic annealing process starting from random configurations. It shouldbe noticed that the Berendsen heat bath does not reproduce rigorously thecanonical ensemble (NVT).

3.1.4 Applications and achievements

As the power of computer has risen, MD simulations have been performedto study fluidic bio and nanosystems with increasing complexity and size [9,112, 21, 49] as illustrated in Figure 3.3, Figure 3.4, and Figure 3.5. Oneof the important benefits of the simulations is that we are able to measure,simultaneously, both macroscopic and microscopic characteristics of the flu-idic processes. Thus, we can measure, on the one hand, the surface tension,the viscosity, the thermal diffusivity and the contact angle and, on the otherhand, the relevant molecular quantities such as the molecular displacementfrequencies and binding energies. MD simulations provide opportunities tovary the properties of the interacting media in ways that are not readilypossible in experiments. By extracting the details of the molecular processesoccurring in the simulations, we also show that the physical interpretationof the parameters is consistent with the model. We believe that this is akey step in the validation of any simulation [9]. Improving models and in-creasing computer power may one day mean that molecular simulations ofwetting will take precedence over experiment. However, in developing ourmodels it is important that we take all necessary steps to ensure that theyreflect the physical world. Many achievements have been accomplished byusing molecular dynamics simulations. Molecular dynamics simulations havebeen performed to analyze biological systems including thousands of atoms,to study nanofluidic systems, to infer macroscale properties and to investi-gate molecular motors. As the capabilities and availability of supercomputersincrease very fast, and the techniques, methods and algorithms are improvedwith increasing velocity, the future of molecular dynamics simulations onlyis limited for our imagination and abilities to extract useful information fromthe usually complex results provided by simulating the physical world whichis complex itself.

In this research, as described in chapter 6 of this thesis, we perform molec-ular dynamics simulations during more than 1 ns of silica-water-air systemsconsisting of more than 800000 atoms using a time step of 2 fs furthermore

28

Figure 3.3: Snapshot from a MD simulation of an aquaporin [21].

29

Figure 3.4: Snapshot from a MD simulation of a carbon nanotubes immersedin Walther [201].

30

Figure 3.5: Snapshot from a MD simulation of the contact angle of a waterdroplet inside a single wall (32, 0) CNT [206].

we perform molecular dynamics simulations during more than 40 ns of someshorter silica-water-air systems consisting of between 15000 and 25000 atomsusing the same time step. By performing this set of simulations we attemptto calibrate a reliable force field for nanofluidics devices. Subsequently, weuse the calibrated force field to study the influence of the air on the watercapillary flow in silica nanochannels. In addition with this set simulations wetry to show that nowadays molecular dynamics simulations can be used asa standard tool to study nanofluidic phenomena in systems including thou-sands of atoms.

31

Chapter 4

Thermophoretic Motion of

Water Nanodroplets Confined

Inside Carbon Nanotubes

Parts of this chapter have been published in: Harvey A. Zambrano, JensH. Walther, Petros Koumoutsakos, and Ivo F. Sbalzarini. Thermophoreticmotion of water nanodroplets confined inside carbon nanotubes. Nano Lett.,9(1):66, 2009.

Abstract

We study the thermophoretic motion of water nanodroplets confined insidecarbon nanotubes using molecular dynamics simulations. We find that thenanodroplets move in the direction opposite the imposed thermal gradientwith a terminal velocity that is linearly proportional to the gradient. Thetranslational motion is associated with a solid body rotation of the waternanodroplet coinciding with the helical symmetry of the carbon nanotube.The thermal diffusion displays a weak dependence on the wetting of thewater-carbon nanotube interface. We introduce the use of the moment scalingspectrum (MSS) in order to determine the characteristics of the motion of thenanoparticles inside the carbon nanotube. The MSS indicates that affinityof the nanodroplet with the walls of the carbon nanotubes is important forthe isothermal diffusion and hence for the Soret coefficient of the system.

32

4.1 Introduction

Carbon nanotubes offer unique possibilities as fluid conduits with applica-tions ranging from molecule separation devices in biocatalysis [112, 137] toencapsulation media for drug storage and delivery [112, 76]. Liquids andsolids in nanochannels may be driven by electrophoresis, osmosis, gradientsin the surface tension also called the Gibbs-Marangoni effect [169], pressuregradients, and thermophoresis. Hence, electrophoresis has been used for driv-ing electrically charged particles in nanosystems [79, 190] and gradients inthe surface tension have been exploited [221] to drive flow through carbonnanotubes (CNTs) immersed into a lipid membrane [53]. Pressure gradientsimposed in nanopipes have been used to generate controlled flows for nano-scale applications [94] and to enhance electrophoretic motion across carbonnanotube membranes. The use of thermal gradients to induce mass trans-port is known as thermophoresis, the Soret effect, or thermodiffusion [62, 64].The first observation of thermodiffusion was reported by Ludwig in 1856 [63],who found differences in samples taken from different parts of a solution inwhich the temperature was not uniform. A systematic investigation of thephenomena was subsequently conducted by Soret [42] in 1879-1881 and byIbbs [100, 12] in 1921-1939 for thermodiffusion in gases. Ibbs found thatthe coefficient of thermal diffusion is more sensitive than any of the othertransport coefficients to the nature of the intermolecular forces [100]. Thus,a complete understanding of the thermal diffusion could provide a power-ful means of investigation of forces between molecules. Although the theo-retical explanation of thermodiffusion for molecules in liquids is still underdebate [33, 157], the investigation of its practical usability is motivated by po-tential applications in nanotechnology. Hence, thermodiffusion was recentlyused as the driving mechanism in artificially fabricated nanomotors [17] asillustrated in Figure 4.1, and thermodiffusion is expected to allow microscalemanipulation and control of flow in nanofluidic devices. In a recent theoret-ical study, thermophoresis was shown to induce motion of solid gold nano-particles confined inside carbon nanotubes [166]. In the present investigation,we study thermophoretic motion of liquid water nanodroplets confined insidecarbon nanotubes.

33

Figure 4.1: Scanning electron microscope (SEM) images of one nanotubebased motor driven by thermal gradients. The CNT works as a nanoconveyortherefore as a thermal gradient is imposed to the system by electric heating,the gold mass is observed to move from its initial position at the center ofthe system until one of the ends of the system [17].

34

4.2 Numerical methods

We perform molecular dynamics (MD) simulations using the MD packageFASTTUBE [198]. This package has been used extensively to study waterconfined inside [206, 223, 118, 113, 8] as illustrated in Figure 3.5 and sur-rounding [200, 203] carbon nanotubes as illustrated in Figure 3.4. The po-tentials governing the water-carbon interaction have been calibrated againstexperiments by considering the contact angle of a water droplet on a graphitesurface [207, 105, 202]. In the present MD simulations we use the rigid SPC/Ewater model [24] and describe the valence forces within the carbon nanotubesusing Morse, harmonic angle, and torsion potentials [198], thus

U (rij, θijk, φijkl) = Kr (ξij − 1)2

+1

2Kθ (cos θijk − cos θc)

2

+1

2Kφ (1− cos 2φijkl) (4.1)

andξij = exp (−γ [rij − rc]) (4.2)

where rij denotes the bond length between two carbon atoms and θ and φare the bending and torsional angles. The Morse stretch and angle bendingparameters were first obtained by Guo et al. [83], these parameters wereoriginally derived to describe the geometry and phonon structure of graphiteand fullerene crystals. The torsion term is needed to provide a measure of thestrain due to curvature of the corresponding graphene sheet. This curvaturestrain avoids the collapse of the carbon nanotube and imparts stiffness withrespect to bending deformations. We include a nonbonded carbon-carbonLennard-Jones potential to describe the van der Waals interaction betweenthe carbon atoms in double-walled nanotubes

Uab(rij) = 4εab

[

(

σab

rij

)12

−(

σab

rij

)6]

(4.3)

where ǫab and σab are the parameters of the potential, here ǫCC = 0.4396 kJmol−1

and σCC = 0.3851 nm. Standard 1-2 and 1-3 nearest neighbor exclusion rulesare applied. This additional Lennard-Jones term is excluded for the simu-lations of single-walled carbon nanotubes unless otherwise specified. The

35

Parameters ValueKr 478.90 kJmol−1

Kθ 562.20 kJmol−1

Kφ 25.12 kJmol−1

ǫCC 0.4396 kJmol−1

σCC 0.3851 kJmol−1

θc 120.00◦

γ 21.867 nm−1

rc 0.1418 nm

Table 4.1: Parameters for the carbon interactions: Kr, rc and γ are theparameters of the Morse potential, Kθ and θc are the angle parameters, Kφ

is the torsion parameter, and ǫcc and σcc are the Lennard-Jones parametersof the carbon-carbon interaction.

parameters of the carbon nanotube potentials are listed in Table 4.1. Thewater-carbon interaction is modeled by a single Lennard-Jones term actingbetween the carbon and the oxygen of the water, consistent with the SPC/Ewater model. For the parameters of the potential, we use a constant σCO

value of 0.3190 nm throughout the simulations but vary the ǫCO parameterin order to investigate the effect of wetting on the thermophoretic motion ofthe water nanodroplet. As a reference value we use ǫCO = 0.3920 kJmol−1,which was found to reproduce the experimental contact angle of 95.3◦ ofa macroscopic water droplet on a graphite surface [206, 207]. We further-more use values of 0.2508 and 0.5643 kJmol−1 corresponding to macroscopiccontact angles of 127.8◦ and 50.7◦, respectively [206, 207]. The nonbondedinteractions, including van der Waals and Coulomb, are truncated at 1.0 nm.The Coulomb potential is smoothed [198, 126] to ensure conservation of en-ergy. For a more detailed description of the potential models and parametersused in this investigation, we refer to Werder et al [207]., Walther et al. [198].and to chapter 3 of this thesis. The equations of motion are integrated intime using the leapfrog scheme [93] with a time step of 2 fs. Periodic bound-ary conditions are imposed in the direction parallel to the tube axis andfree space conditions in the normal directions. We equilibrate the systemfor 0.1 ns. During the equilibration we couple the water and the CNT totwo separate Berendsen thermostats [25] with a time constant of 0.1 ps. Af-ter equilibration we impose a thermal gradient along the axis of the CNT.

36

Figure 4.2: Snapshot from a molecular dynamics simulation of a water nano-droplet confined inside a carbon nanotube.

The heated zones are two 2 nm sections at the ends of the computationalbox as illustrated in Figure 4.2. We study the effects of thermal gradients(∇T) ranging from 0.40 to 3.68 Knm−1. We note that the lower gradients(∇T < 1.0Knm−1) are attainable experimentally [17] whereas the largergradients help us assess trends and to increase the signal-to-noise ratio inour simulations. We measure the position of the center of mass of the waternanodroplets (zCOM) and the corresponding center of mass velocity (vCOM)during the simulation. Moreover, we measure the radial profiles of the den-sity and the axial and tangential velocity of the water nanodroplet. Theprofiles are sampled in the central region of the droplet and exclude the freesurface at the droplet caps. The resolution of the binning is 0.117 nm for theradial profiles. The coefficient of thermal diffusion (Dt) is estimated from

Dt =νCOM

∇T(4.4)

the Soret coefficient is defined as [62, 64, 33, 151]

St =Dt

D(4.5)

where D is the isothermal diffusion coefficient. We determine D from themoment scaling spectra (MSS) [68, 66, 164] with a temporal resolution of 2 ps.Each trajectory is represented as a sequence zCOM(n) of the n = 1, . . . ,Mposition of the center of mass over time. The time difference between twopoints is given by δt = ∆n ∗ 2 ps. Our analysis is based on computing thefirst 6 moments of displacement

µν (∆n) =1

M −∆n

M−∆n∑

n=1

|zCOM (n+∆n)− zCOM (n)|ν (4.6)

37

for ν = 0, . . . , 6 and ∆n = 1, . . . ,M/3. This includes the classical meansquare displacement (MSD) as the special case where ν = 2. We assumeeach moment µ to depend on the time shift δt = ∆n∗2 ps in a power law [68]

µν (δt) = 2dDν (δt)γν (4.7)

where d = 1 is the dimension of the space. The scaling coefficients γν arethus determined by linear least-squares regression of log(µnu) vs log(δt). Forν = 2 we obtain the isothermal diffusion coefficient D = D2 from the yaxis intercept y0 as D = 0.5 × 10y0 . The plot of γnu vs ν is called themoment scaling spectrum (MSS) [68]; for all strongly selfsimilar diffusionprocesses, the MSS shows a straight line through the origin. The slope SMSS

of this line is determined using linear least-squares regression. SMSS = 0.5indicates pure normal diffusion, SMSS = 1.0 is characteristic for ballisticmotion. Superdiffusive processes have 0.5 < SMSS < 1.0, and subdiffusion0.0 < SMSS < 0.5. MSS analysis is robust against noise in the trajectoryand allows more accurate regression due to the almost perfect linearity ofthe MSS and the fact that SMSS has a smaller estimator variance than γ2.In addition it enables classification of different motion types with uniformsensitivity along the trajectory.

4.3 Results

In this investigation we perform molecular dynamics simulations of a waternanodroplet confined inside a carbon nanotube (Figure 4.2) subject to a ther-mal gradient. We use a zigzag carbon nanotube with a chiral vector of (30, 0)which corresponds to a radius of 1.17 nm and a pitch angle of helical symme-try (αzz) of ±60◦ [165, 20]. The size of the computational domain along theaxis of the carbon nanotube is 42.6 nm which requires 12000 carbon atoms.The water droplet consists of 514 water molecules and the confinement resultsin a droplet mean radius of approximately 0.95 nm and length 6.00 nm; cf.Figure 4.2. We impose thermal gradients of 0.00, 0.40, 0.52, 0.79, 1.05, 1.32,1.58, 1.71, 1.97, 2.10, 2.63, 2.89, 3.16, and 3.68Knm−1. We observe, for thenonzero gradients, a thermophoretic motion of the water nanodroplets in thedirection opposite the thermal gradient (Figure 4.3). Moreover, the centerof mass velocity of the water nanodroplet increases as higher gradients areimposed. The mean terminal velocity of the nanodroplet is approximately22m s−1 for a thermal gradient of 1.05Knm−1, cf. Figure 4.4, and similar in

38

0

5

10

15

20

25

30

35

40

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

z CO

M (

nm)

time (ns)

Figure 4.3: Position of the center of mass of the nanodroplet zCOM as afunction of the time for three different thermal gradients (red, 1.97 Knm−1;green, 1.58 Knm−1 and blue, 1.05 Knm−1).

39

-40

-20

0

20

40

60

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

v CO

M (

nm n

s-1)

time (ns)

Figure 4.4: Mean velocity of the center of mass of the nanodroplet (νCOM)as a function of the time for the case with thermal gradient of 1.05 Knm−1.

40

0

5

10

15

20

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

v z (

nm n

s-1)

r (nm)

Figure 4.5: Time average axial velocity profile (vz) of a water nanodropletconfined inside single-walled carbon nanotubes subject to a thermal gradientof 1.05Knm−1.

magnitude to the terminal velocity of 25m s−1 found in the previous studyof thermophoretic motion of gold nanoparticles confined inside carbon nano-tubes [165]. The axial velocity profile (νz) of the water nanodroplets displaysa maximum value in the vicinity of the fluid-solid interface (Figure 4.5).Hence the fluid motion of the water nanodroplets exhibits recirculation ina frame of reference fixed to the center of mass of the nanodroplet. Thecorresponding tangential velocity profile (vt) shown in Figure 4.6 displaysa solid body rotation similar to the rotation of the solid gold nanoparticleobserved in our previous study [165]. The motion is associated with largetemporal fluctuations, cf. Figure 4.4 and Figure 4.7, which are related tothe change in the direction of rotation; cf. Figure 4.7. Moreover, the angleof rotation appears to be correlated to the two possible paths imposed bythe chirality vector of the carbon nanotubes of ± 60◦ [165]. From the sim-ulations (Figure 4.7, 4.8) we find an angle between the maximum value ofthe tangential velocity, measured in the vicinity of the surface of the car-

41

-10

-5

0

5

10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

v t (

nm n

s-1)

r (nm)

Figure 4.6: Time average tangential velocity (νt) profile of a water nano-droplet confined inside single-walled carbon nanotubes subject to a thermalgradient of 1.05Knm−1: the red line represents the measured velocity profileand the green line the best fit to a solid body rotation.

42

-60

-40

-20

0

20

40

60

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

v t (

nm n

s-1)

r (nm)

Figure 4.7: Tangential velocity profiles of a water nanodroplet confined insidesingle-walled carbon nanotubes subject to a thermal gradient of 1.05Knm−1

for different time intervals (red line, 0.56–0.70 ns; green line, 0.70–0.92 ns;blue line, 0.92–1.04 ns; pink line, 1.04–1.20 ns).

bon nanotube, and the axial velocity of the nanodroplet, has a mean valueof approximately 67◦. Thus, the rotation of the nanodroplet is determinedby the helical symmetry of the CNT. In order to determine the isothermaldiffusion coefficient, we carry out long (200 ns) simulations of the water nano-droplet confined inside the carbon nanotube with zero temperature gradient.Below time scales of about 200 ps, the scaling of the moments of displace-ment indicates ballistic motion (see Figure 4.9a) with SMSS = 0.97. Thescaling plot then shows a pronounced kink, with a long time scaling thatis characteristic for diffusive motion (see MSS plot in Figure 4.9b). In thisdiffusive regime we find an isothermal diffusion coefficient of 8.8 nm2 ns−1

with SMSS = 0.46, indicating pure normal diffusion. This value is in rea-sonable agreement with values previous obtained for water under nanoscaleconfinement (D = 0.94 − 5.7 nm2 ns−1) [179, 193, 162]. For the simulations

43

0

10

20

30

40

50

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

v z (

nm n

s-1)

r (nm)

Figure 4.8: Axial velocity profiles of a water nanodroplet confined insidesingle-walled carbon nanotubes subject to a thermal gradient of 1.05Knm−1

for different time intervals (red line, 0.56–0.70 ns; green line, 0.70–0.92 ns;blue line, 0.92–1.04 ns; pink line, 1.04–1.20 ns).

44

with a nonzero thermal gradient, we observe a linear increase in the cen-ter of mass velocity as function of the thermal gradient (Figure 4.10) whichconfirms that the motion is thermophoretic. We expect that the motion isrelated to high-frequency phonon vibrations of the carbon nanotubes, i.e.,the kinetic pressure of the carbon nanotubes as demonstrated by Schoenet al. [166]. We note that the present classical formulation will excite allphonons simultaneously, whereas a full quantum description will suppresshigh-frequency phonons at low temperatures. Thus, the speed of the nano-droplets may be quantized at low temperatures. The extracted coefficientof thermal diffusion is 20.9 nm2( ns K)−1, which corresponds to a Soret co-efficient of 2.37K−1. This value is 3 orders of magnitude larger than thecoefficients reported by Platten et al. [64] and Saghir et al. [162] for mixturesof water-methanol and by Reith and Muller-Plathe [158] of binary Lennard-Jones liquids. In order to evaluate the influence of the wetting of the water-carbon interface on the thermophoretic motion of the nanodroplet, we varythe Lennard-Jones parameter ǫCO, from the reference value of 0.3920 kJmol−1

to 0.5643 and 0.2508 kJmol−1, which correspond to strongly hydrophilic andstrongly hydrophobic interfaces, respectively. The resulting water densityprofiles display the characteristic structure at the interface with enhancedstructure for increasing hydrophobicity (Figure 4.11). We observe a smallbut systematic change in the center of mass velocity of the water nano-droplet, cf. Figure 4.10, and the coefficients of thermal diffusion vary fromthe reference value of 20.9 nm2( nsK)−1 to 18.8 nm2( nsK)−1 for the stronglyhydrophilic interface to 23.1 nm2( nsK)−1 for the strongly hydrophobic in-terface. This behavior is consistent with the increased surface adhesion andreduced isothermal diffusion for the hydrophilic interface. Conversely, thesuperhydrophobic interface exhibits a reduced adhesion and hence increasedisothermal diffusion as shown in Figure 4.10. To estimate the effect of wettingon the Soret coefficient, we repeat the long (200 ns) isothermal simulations forthe system setting with different ǫCO values. From these simulations we mea-sure an isothermal diffusion coefficient of 20.8 nm2 ns−1 (SMSS = 0.46) and7.9 nm2 ns−1 (SMSS = 0.54) for ǫCO values of 0.2508 and 0.5643 kJmol−1,respectively. The corresponding Soret coefficients are ST = 0.90K−1 andST = 2.92K−1. Hence, the hydrophilic interface exhibits a higher Soretcoefficient than the hydrophobic interfaces, mainly caused by large changein the isothermal diffusion. We finally consider the thermophoretic motionof a water nanodroplet confined inside single carbon nanotubes of differ-ent chirality and inside double-walled carbon nanotubes. The additional

45

Figure 4.9: (a) Moments of displacement for the simulations using εCO =0.3920 kJmol−1. Solid blue lines show the temporal scaling of the first sixmoments of displacement. The least-squares fits are indicated by red dashedlines. The scaling coefficients are given by the slopes of the dashed lines.On time scales below 200 ps, the scaling corresponds to ballistic motion.Diffusive behavior with D = 8.8 nm2 ns−1 is recovered for longer time scales.(b) Moment scaling spectrum. The circles mark the scaling coefficients ascomputed from the fits in (a); the solid blue line indicates the linear least-squares fit with a slope of SMSS = 0.46. The red dashed lines of slopes 0.5and 1.0 show the theoretically expected spectra for pure normal diffusion andballistic motion, respectively.

46

0

20

40

60

80

100

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

v CO

M (

nm n

s-1)

thermal gradient (K nm-1)

Figure 4.10: Influence of wetting effect on the center of mass mean velocityof a water nanodroplet confided inside a zigzag carbon nanotube. Meanvelocity of the center of mass VCOM as a function of the imposed thermalgradient for three different ǫCO values (red line, ǫCO = 0.5643 kJmol−1; greenline, ǫCO = 0.3920 kJmol−1; and blue line, ǫCO = 0.2508 kJmol−1). Red (∗),green (×), and blue (+) represent the particular mean velocities of each casesimulated. A black square represents the mean velocity of the center of massof a nanodroplet confined inside a double-walled carbon nanotube with animposed thermal gradient of 2.83Knm−1.

47

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

dens

ity (

g cm

-3)

r (nm)

Figure 4.11: Influence of wetting effect on the density profile of a water nano-droplet confided inside a zigzag carbon nanotube. Radial density profile forthe three different ǫCO values (red line, ǫCO = 0.2508 kJmol−1; green line,ǫCO = 0.3920 kJmol−1; blue line, ǫCO = 0.5643 kJmol−1).

48

0

50

100

150

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

v CO

M (

nm n

s-1)

thermal gradient (K nm-1)

Figure 4.12: Influence of chirality on the center of mass mean velocity of awater nanodroplet confined inside a carbon nanotube. Chirality: Red (+)(17, 17), green (×) (20, 14), and blue (∗) (30, 0).

single-walled carbon nanotubes include (17, 17) armchair and (24, 14) chi-ral carbon nanotubes, both with a diameter similar to the zigzag (30, 0)carbon nanotubes previously considered. For all the different chiralities weobserve a consistent thermophoretic motion of the water nanodroplets asshown in Figure 4.12. Moreover, we find that the nanodroplet moves withthe highest velocity inside the (17, 17) armchair carbon nanotubes and withthe lowest velocity inside the zigzag (30, 0) carbon nanotubes. These resultsare in agreement with recent molecular dynamics and quantum chemistrystudies of self-diffusion of water confined inside armchair and zigzag car-bon nanotubes [128]. For the double walled carbon nanotubes, we impose athermal gradient of 2.83Knm−1 and observe a thermophoretic motion (solidblack square in Figure 4.10) similar to that observed for single-walled car-bon nanotubes. The terminal center of mass velocity is 53m s−1 and similarto the velocity observed for the hydrophilic (ǫCO = 0.5643 kJmol−1) single-

49

0

50

100

150

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

v CO

M (

nm n

s-1)

thermal gradient (K nm-1)

non-periodicperiodic

Figure 4.13: Influence of periodicity on the MD simulations of a systemconsisting of a water droplet inside a carbon nanotube.

walled carbon nanotube of 55 nmns−1. This behavior is consistent with theenhanced water-carbon interaction due to the presence of the second layerof carbon atoms. In order to observe any influence of the imposed periodicboundary conditions, we have performed simulations of a water droplet con-fined inside a zig-zag (30, 0) carbon nanotube removing the periodicity of thesystem and imposing different thermal gradients. In Figure 4.13 a very sim-ilar qualitative behaviour of the center of mass velocity of the water dropletis observed for the periodic and non-periodic system (In chapter 3 of thisthesis is included a description about artifacts induced by imposing periodicboundary conditions to a system).

4.4 Conclusions

The present molecular dynamics simulations have demonstrated that waternanodroplets may be driven through single- and double-walled carbon nano-

50

tubes by imposing a thermal gradient along the axis of the carbon nano-tube. The driving force is thermophoretic, and the resulting thermal diffusioninvolves both a translational and rotational motion of the nanodroplets. Therotation is imposed by the helical symmetry of the carbon nanotubes. Wefind a weak influence of wetting on thermophoresis, but a strong influenceon the isothermal diffusion and hence the Soret coefficient. In summary,thermophoresis may be a viable mechanism for controlling the motion ofwater nanodroplets confined inside a single- and double-walled carbon nano-tubes.

51

Chapter 5

Carbon Nanotubes Based

Molecular Motors Driven by

Thermophoresis

Parts of this chapter has been published in: H. A. Zambrano, J. H. Walther,and R. L. Jaffe. Thermally driven molecular linear motors: A moleculardynamics study. J. Chem. Phys.,131:241104, 2009.

Abstract

We conduct MD simulations of the molecular linear motor recently presentedby Somada et al. [Nano. Lett., 9, 62, (2009)]. The system consists of a coax-ial carbon nanotube with a long outer carbon nanotube confining and guid-ing the motion of an inner short, capsule-like nanotube. Somada et al. usedTransmission Electron Microscopy (TEM) to visualize the motion of the cap-sule and argued that the motion is driven by thermal energy. The present MDsimulations indicate that the motion of the capsule may be strongly influ-enced by thermophoretic forces induced by thermal gradients e.g., imposed bythe TEM. The simulations find large terminal velocities of 100–400 nmns−1

for imposed thermal gradients in the range 1–3Knm−1. Moreover, the resultsindicate that the thermophoretic force is velocity dependent and decreasesfor increasing velocity.

52

5.1 Introduction

In a recent experimental study, Somada et al. [177]. fabricated a molecularlinear motor system consisting of co-axial carbon nanotubes (CNT) as shownin Figure 5.1. In the system an inner CNT is found to move throughout thein vacuo internal space of an outer CNT, working as a molecular linear mo-tor. The experimental setup consists of a capped capsule-like short carbonnanotube with a chiral vector of (12, 0) encapsulated in the interior hollowspace of a single wall CNT with a chiral vector of (22, 0). The hollow space isapproximately 8.5 nm long and limited by two fixed capped CNTs with a chi-ral vector of (12,0). The total length of the outer CNT is more than 100 nmlong. Using Transmission Electron Microscopy (TEM), Somada et al. [177].find that the inner CNT (capsule) changes its position including stop eventsat the two ends of the hollow space. During the 170 s long experiment, thecapsule is found to travel back and forth seven times. However the motionof the capsule is observed indirectly due to the low time resolution of theTEM of 0.5 s. The picture obtained from the TEM experiment furthermoreindicate that the capsule is undergoing rotation during the translational mo-tion. Somada et al. [177]. propose that the mechanism driving the capsuleis related to thermal activation energy which is in equilibrium with the vander Waals (vdW) energy gain due to the interaction between the caps ofthe carbon nanotubes. Since the theoretical activation time is shorter thanthe activation time measured in the experiments. Somada and coworkersinclude an additional effect contributing to the total friction force, which isassociated with a deformation caused by thermal fluctuation of the graphiticlattices of the outer CNT. They argue that the thermal energy not onlyactivates the capsule motion but also obstructs its travel by deforming thehollow space of the system [177]. In order to employ the concept of a molecu-lar linear nanomotor with potential applications such as mass nanotransportsystems [156, 180, 76, 67, 172], and archival memory devices [19] a detailedexplanation of the driving mechanism of the capsule should be addressed. Inthis study, we propose that the motion of the capsule observed by Somadaet al. [177] may be influenced or directly driven by thermophoretic forces in-duced by the thermal gradient imposed by the electron beam irradiated fromthe TEM. In experimental and theoretical studies Howe et al. [98], Yokotaet al. [212], Wang et al. [204] and Biskupek et al. [27] found that electronbeam irratiation generates a temperature increase in the irradiated regiondue to electron thermal spikes, which are generated as incident electrons

53

Figure 5.1: (a-g) A series of TEM images showing linear motion of a CNTcapsule. The exposure time for each image is 500 ms, and the total recordingtime was 10 min. The CNT capsule is 0.95 nm in diameter and 3.2 nm inlength. The diameter of the host SWNT is 1.6 nm, and the length of thehollow space is about 8.5 nm. The CNT capsule is encapsulated in a hollowspace between A and B as illustrated in (h). (i) Capturing intervals at sidesA and B for seven laps [177].

54

Figure 5.2: Schematic of the computational setup. Cross-sectional view ofthe system, the outer CNT is a (22,0) zigzag CNT and the inner one is a(12,0) zigzag CNT. A thermal gradient is imposed by heating the end sections(in gray) of the outer CNT.

pass through and/or collide with the irradiated system. Moreover, inducedeffects such as defect production, annealing and heating by electron-phononcoupling have been investigated by Krasheninnikov [120, 119]. Using Molecu-lar dynamics simulations, Kresheninnikov [119] argues that in carbon nanos-tructures the kinetic energy of an incident electron beam is converted intothermal energy resulting in an increase in temperature higher than expectedfor bulk systems due to the energy dissipation is affected by the small systemsize in one or more directions. In this work we perform molecular dynamicssimulations in order to demonstrate that the motion of the inner CNT, in asystem similar to the studied by Somada et al. [177]. is strongly influencedby thermophoretic forces. We show quantitatively that a thermal gradientas small as 1.2 K nm−1 could be the origin of the motion of the capsule [17].For our simulations we use the MD package FASTTUBE [198], which hasbeen used extensively to study thermophoretic motion of liquids and solidsconfined inside single and double wall CNTs [215, 172, 167, 166, 165].

5.2 Methodology and thermodiffusion study

We simulate the system studied by Somada et al. [177]. using a double wallcarbon nanotube system as illustrated in Figure 5.2. The system consists ofan outer 42.6 nm long carbon nanotube with a chiral vector of (22, 0) corre-

55

sponding to a diameter of 1.723 nm. The inner capsule is modeled as an openshort 3.195 nm long carbon nanotube with a chiral vector of (12, 0), and di-ameter 0.94 nm. We describe the valence forces within the CNT using Morse,harmonic angle and torsion potentials [198]. We include a nonbonded carbon-carbon Lennard-Jones potential with parameters ǫCC = 0.4396 kJmol−1 andσCC = 0.3851 nm to describe the vdW interaction between the carbon atomswithin the double wall portion of the system. We equilibrate the system at300K for 0.1 ns. During the equilibration we couple the system to a Berend-sen thermostat with a time constant of 0.1 ps. After the equilibration weimpose thermal gradients in the range of 0.00-4.20 Knm−1, which are im-posed by heating two zones at the ends of the outer CNT as illustrated inFigure 5.2. We measure the position of the center of mass of the inner CNTduring the simulation. We observe, for gradients of 1.18 Knm−1 or higher,a directed motion of the capsule in the direction opposite to the imposedthermal gradient as shown in Figure 5.3a. For a thermal gradient of 1.18Knm−1 the mean terminal velocity is approximately 170.0 nmns−1, whichis higher than the velocity measured in our previous studies on thermophore-sis of water nanodroplets and gold nanoparticles confined inside carbon nan-otubes [215, 165] but similar to the velocity measured in the simulations byBarreiro et al. [17] (Figure 4.1). Moreover, we find a consistent increase inthe therminal velocity for increasing thermal gradients (see Figure 5.3). Weobserve from the time history of the COM position and velocity (Figure 5.3)that before the capsule has started a directed motion, oscilations of the COMposition are evident. We believe that the observed oscilations are related tothe static friction mainly owing to the energy barrier between the two coaxialnanotubes [17]. We note that as a higher thermal gradient is imposed thetime period in which the COM position is oscilating get reduced.

5.3 Friction and thermophoretic force study

To confirm that the motion of the capsule is driven by thermophoresis we per-form additional simulations in order to study the friction and thermophoreticforces acting on the inner CNT. We carry out a force analysis similar to thatperformed by Schoen et al. [165]. In these simulations, we constrain the ve-locity of center of mass of the inner CNT. We extract from the simulationsthe external forces required to drive the inner CNT for different constrainedvelocities and different imposed thermal gradients (Figure 5.4). To measure

56

10

15

20

25

30

35

40

1.3 1.4 1.5 1.6 1.7 1.8

CO

M p

ositi

on (

nm)

time (ns) (a)

-100

0

100

200

300

400

500

1.3 1.4 1.5 1.6 1.7 1.8

CO

M v

eloc

ity (

nm/n

s)

time (ns)

-100

0

100

200

300

400

500

1.3 1.4 1.5 1.6 1.7 1.8

CO

M v

eloc

ity (

nm/n

s)

time (ns) (b)

Figure 5.3: Center of mass position (COM) (a) and velocity (b) as a functionof time for three different thermal gradients (blue (*), 3.16 K nm−1; green(×), 1.58 K nm−1 and red (+), 1.18 Knm−1).

57

-100

-80

-60

-40

-20

0

0 1 2 3 4

forc

e (k

J/m

ol n

m)

thermal gradient (K/nm)

-100

-80

-60

-40

-20

0

0 1 2 3 4

forc

e (k

J/m

ol n

m)

thermal gradient (K/nm) (a)

-60

-50

-40

-30

-20

-10

0

10

1 10 100 1000

forc

e (k

J/m

ol n

m)

COM velocity (nm/ns) (b)

Figure 5.4: External force acting on the constrained inner CNT (a) externalforce as a function of the imposed thermal gradient for different constrainedvelocities (blue line and (+), 4 nmns−1; red line and (×), 16 nmns−1. (b)External force acting as a function of the center of mass (COM) velocityfor different thermal gradients (red (+), 0.0 K nm−1; green (×), 1.0 K nm−1;blue (*), 2.0 K nm−1; and fuchsia (squares), 3.0 K nm−1)

58

the isothermal friction of the system we imposed an isothermal constraint(thermal gradient of 0.0 K nm−1) while we vary the constrained COM ve-locity. At non-zero thermal gradients we measure the combined friction andthermophoretic forces; a positive force indicates a resistance to the motion,whereas a negative force is indicative of thermophoresis. In Figure 5.4a weshow the mean external force as a function of the imposed thermal gradientfor two constrained COM velocites, the external forces are subject to largefluctuations represented by the error bars in the corresponding function. InFigure 5.4b we show the external driving forces as a function of the COM ve-locity for different thermal gradients (0.0, 1.0, 2.0 and 3.0 K nm−1), extractedfrom more than eighty individual simulations. We find a systematic increaseof the thermophoretic force as a higher thermal gradient is imposed to thesystem (Figure 5.4). Furthermore, we measure an isothermal friction thatis small compared to the measured thermophoretic force as shown in Fig-ure 5.3. We infer, from the simulations, a reduction of the thermophoreticforce as a higher velocity is imposed to the inner CNT (Figure 5.4). Weconjecture that the driving thermophoretic force is velocity dependent thusthe thermophoretic force decreases when a higher velocity is reached. Webelieve that, for different imposed thermal gradients, the corresponding ter-minal velocity is governed by the velocity dependence of the thermophoreticforce velocity rather than a match between the thermophoretic force (mea-sured at zero velocity) and the static friction. We observe from Figure 5.4bthat the zero external force is obtained at approximately 100 nmns−1 for 1K nm−1, at 250 nmns−1 for 2 K nm−1 and at 400 nmns−1 for 3 K nm−1 inreasonable agreement with the terminal velocities observed in Fig. 5.3b.

5.4 Conclusions

In summary, we propose that a thermal gradient imposed due to the par-tial irradiation of the system by the electron beam of the microscope usedin the experiments [177], could be the main mechanism responsible of themotion of the inner CNT. We believe that the thermal activation energyproposed by Somada et al. [177] as driven mechanism could be part of themechanism to extricate the capsule from the energy minima at the endsof the hollow space. However, we conjecture that since the capsule hasstarted its motion the mechanism driving the capsule to the opposite hollowspace end is directly influenced by thermophoretic forces. We believe that

59

our hypothesis contribute to the utilization of thermophoresis based linearnanomotors with several applications in transport and delivery of substancesencapsulated in carbon nanostructures confined inside a host carbon nan-otube [92, 91, 15, 213, 41, 75, 133]. Moreover, with this Note we hope toencourage more experimental work to study thermophoretic linear nanomo-tors and its potential applications.

60

Chapter 6

Molecular dynamics study of

the interface between silica,

water and air

Parts of this chapter have been included in the following papers: H. A.Zambrano, J. H. Walther and R. L. Jaffe. Molecular Dynamics Simulationsof the Interface Between Silica and Water at High Air Pressures. J. Mol.Liquids, submitted.

Abstract

In this study we present a new force field suitable for performing moleculardynamics simulations of amorphous silica and water at different air pres-sures. We calibrate the interactions of each specie presents in the systemusing dedicated criteria such as the contact angle of a water droplet on asilica surface and the gas solubility in water at different pressures. Usingthe calibrated force field, we perform MD simulations to study the interfacebetween a hydrophilic silica subtrate and water surrounded by air at differ-ent pressures. We find the static water contact angle to be independent ofthe air pressure imposed on the system. Our simulations reveal that a layerwith higher concentration of gas exists close to the silica surface. We believethat the presence of a nano-layer of gas at the water-slica interface couldpromote the nucleation and stabilization of surface nanobubbles. Finally, weperform simulations to investigate the water capillary filling of nanochannels

61

with heights below 10 nm at different air pressures; specifically, we analyzethe earliest stages of the filling process. The simulations show that the pres-ence of air could affect the capillary filling process by modifying the dynamiccontact angle of the advancing meniscus reducing the capillary filling speed.

6.1 Introduction

The static and dynamic behavior of fluids at a solid substrate has been sub-ject of intense research during the last decades [217, 146, 61, 99, 48, 163, 105].Wetting phenomena has been extensively studied [51, 28], but several ques-tions relating to interfacial dynamics properties remain open, due to theoreti-cal, experimental or computational limitations. Wetting is essential and ubiq-uitous in a variety of natural and technological processes [153, 196, 178, 222].Miniaturization, a trend in several technological fields, is leading to complexstructures on the nanoscale [1, 29, 222]. Development of a new generationof nanofluidic devices will require a comprehensive understanding of wet-ting phenomena at the nanoscale. Silicon dioxides-water systems interactingwith air are abundant in nature and play fundamental roles in a diversityof novel science and engineering activities such as silicon based nanosen-sor devices, nanoscale lab-on-a-chip systems and DNA microarrays technolo-gies [86, 88, 1, 54, 174]. Although extensive experimental, theoretical andcomputational work has been devoted to study the nature of the interactionbetween silica and water [123, 57, 195, 60, 87, 69, 187, 144, 196, 44, 49], manyfundamental questions are, currently, subject of intense debate. Recently, in anumber of different experimental investigations, systematic deviations in thefilling rate from classical predictions have been measured for channels withnanoscale dimensions [145, 147, 184, 182, 132, 148]. Thamdrup et al. [184] at-tribute the measured deviations to the presence of gas nanobubbles. The ex-istence of nanoscale gas bubbles at solid-liquid interfaces have been proposedas an explanation for the long range hydrophobic forces measured in surfaceforce apparatus experiments [104, 13, 146]. Subsequently, nanobubbles havebeen observed and inferred in a variety of experimental and theoretical workson hydrophobic [217, 13, 103, 107, 14, 31, 220, 218, 143, 34, 171, 130] and hy-drophilic surfaces [217, 38, 130]. Nevertheless, the long life time of interfacialnanobubbles contradicts the classical theory of nanoscale bubbles stabilitybecause their large Laplace pressure would cause a rapid diffusive out-flux ofgas [65, 129]. However, in recent theoretical and experimental studies some

62

potential stabilization mechanisms have been suggested to explain the ob-servations of long lived nanobubbles [72, 138, 220, 219, 34, 55, 59]. In thiswork we parametrize and calibrate interaction potentials in order to conductlong Molecular Dynamics (MD) simulations of systems including silica, waterand air. Specifically, we study the role of air on the wetting of water-silicainterfaces and the effect of the presence of air on the capillary filling speedof water in silica nanochannels.

6.2 Methodology and force fields

We conduct long MD simulations of hydrophilic silica-water-air systems us-ing the MD package FASTTUBE [198]. FASTTUBE has been used exten-sively to study water inside and surrounding carbon nanotubes, and nanode-vices [206, 203, 202, 215, 214]. Since the objective of this work is the study ofnanofluidic interfacial phenomena we need an effective silica-water-air modelsuitable to allow MD simulations of large systems in the range of nanosecondsand reliable enough to reproduce the relevant effects related to the wettingproperties of nanoscale systems. We use well tested models [192, 24] forthe individual species present in our systems. Nevetheless, we carry out aprocess of parametrization and calibration using dedicated criteria for eachinteraction involved in the system [32]. The specific criteria used to calibratethe models are relevant to the properties we seek to reproduce.

6.2.1 Silica interaction potential

Amorphous silica is described by the TTAMm potential model developed byGuissani et al. [82], which is a modification of the TTAM potential modeldeveloped by Tsuneyuki et at. [192]. The TTAMm model includes a Lennard-Jones 6-18 potential and a Buckingham potential

φab(rij) =qaqb

4πε0r2ij+ αab exp

(−rijρab

)

− Cab

r6ij

+4ǫab

[

(

σab

rij

)18

−(

σab

rij

)6]

, (6.1)

where qa and qb represent the partial charges of the atomic species a and b,ε0 the vacuum permittivity, rij the inter-atomic distance, σab the distance at

63

Parameter Value

ǫSi−Si 1.277× 103 kJmol−1

σSi−Si 4.00× 10−2 nm

ǫOs−Os 4.60× 10−2 kJmol−1

σOs−Os 0.220 nm

ǫSi−Os 1.083 kJmol−1

σSi−Os 0.130 nm

αSi−Si 8.417× 1010 kJmol−1

ρSi−Si 1.522× 102 nm

CSi−Si 2.240× 10−3 kJmol−1nm6

αOs−Os 1.696× 105 kJmol−1

ρOs−Os 0.283× 102 nm

COs−Os 0.0207 kJmol−1nm6

αSi−Os 1.0347× 106 kJmol−1

ρSi−Os 0.480× 102 nm

CSi−Os 6.825× 10−3 kJmol−1nm6

qSi 2.400eqOs −1.200e

Table 6.1: Lennard-Jones, Buckingham parameters and partial charges forthe TTAMm model.

which the inter-particle LJ potential is zero, ǫab the depth of the LJ potentialwell, αab, ρab and Cab represent the Buckingham force field parameters. Aoverview of the potential parameters is listed in Table 6.1, where the sub-scripts Si and Os represent, respectively, silicon and oxygen atoms in silica.To describe the electrostatic interactions in the bulk silica system we em-ploy partial charges obtained from the TTAM model [192], qSi = +2.4e andqOs = −1.2e. In order to reproduce amorphous silica systems, we replicatea crystoballite cell to build a slab. Subsequently, we perform MD simula-tions imposing periodic boundary conditions in X and Y directions, whileimposing non-periodic boundary conditions in Z-direction in order to createa free surface. We use an integrating time step of 1 fs and follow a similarannealing procedure as employed by Cruz-Chu and et al. [49]. Therefore, wecouple the system to a Berendsen [25] heat bath, heating the crystoballiteslab to 3000K during 10 ps, subsequently, quenching the system from 3000Kto 300K by using a cooling rate of 70Kps−1 until the equilibrium state isreached. Once silica is equilibrated, we classify the atoms at the surface ofthe silica slab according to their connectivity. Therefore a pair of atoms sepa-rated for a distance shorter than 0.2 nm are considered as covalently bonded.

64

Parameter Value

ǫOw−Ow 0.6501 kJmol−1

σOw−Ow 0.3166 nm

qOw -0.8476e

qHw 0.4238e

Table 6.2: Lennard-Jones parameters and partial charges for the SPC/Ewater model [24].

Silicon atoms with less than four covalent bonds and oxygen atoms with lessthan two covalent bonds are considered as dangling atoms. We measure aconcentration of dangling oxygens of 1.5 atoms per nm2 and a concentrationof dangling silicons of 1.05 atoms per nm.

6.2.2 Water interaction potential

Water is described by the rigid extended simple point charge potential SPC/Emodel [24]. It consists of a smoothly truncated Coulomb potential [126]acting between the partial charges on the oxygen (qOw = −0.8476e) andhydrogen (qHw = +0.4238e) atoms and a Lennard-Jones 12-6 potential [110]as shown in Eq. (6.2). The potential parameters for the water model arelisted in Table 6.2.

Uab(rij) = 4εab

[

(

σab

rij

)12

−(

σab

rij

)6]

. (6.2)

6.2.3 Silica-water interaction potential

Since long simulation times (more than 20 ns) are required to study nanoflu-idic systems, our objective is to derive interaction potential models to beboth highly efficient in terms of processing time and sufficient accurate toreproduce reliably the macroscale properties of the silica-water interface. Ina previous work, Hassanali and Singer [87] developed a model which is anextension of the van Beest, Kramer, and van Santen (BKS) model for silicaand of the SPC/E model [24] for water. They added three body interactionterms which are not found in the standard BKS model and obtained thepotential parameters from ab initio quantum chemical studies on small frag-ments of silica-water systems. In the present study we retain the simple and

65

Figure 6.1: Contact angle of a water nanodroplet on an amorphous silicaslab. Snapshot from a MD simulation of a water nanodroplet mounted on asilica slab subject to a high air pressure (10 bar).

efficient two body Born-Huggins-Mayer potential, which consists of Coulombpotential and Buckingham potential terms

µab(rij) =qaqb

4πε0r2ij+ αab exp

(−rijρab

)

− Cab

r6ij, (6.3)

where qa and qb represent the partial charges of the atomic species a and brespectively, ε0 the vacuum permittivity, rij the inter-atomic distance and,αab, ρab and Cab represent the Buckingham force field parameters. As a start-ing point, we use some Buckingham parameter Cab determined by Hassanaliand Singer [87]. Due to the fact that, for amorphous silica, the surfacialelectrostatics is significantly different to the bulk electrostatics, we modifythe partial charges of the rigid silica model to follow a procedure similar tothe employed by Cruz-Chu et al [49]. The partial charges values (we use)are taken from the soft potential developed by Takada et al. [181], which haselectrostatic interactions weaker than the partial charges used in classicalsilica models thus qOs = −0.65e for the oxygen atoms and qSi = 1.3e for thesilicon atoms. We equilibrate the silica slab as described in subsection 2.1.

66

The water is immobilized during the amorphous silica equilibration and cool-ing process. Subsequently, we equilibrate the water, immobilizing the silicaand coupling the water to a Berendsen heat bath [25] until the energy ofthe system has reached a constant value. We perform simulations of a waterdroplet of 18000 molecules placed on a silica slab of 37.92 nm × 37.92 nm ×2.5 nm (ca 350000 atoms) as shown in Figure 6.1. For the simulations we usean integrating time steps of 2 fs. We notice that the size of the silica surfaceand the water droplet we study are much longer that the systems previouslysimulated by Cruz-Chu et al. [49] and, Hassanali and Singer [87]. The sizeof the silica-water system is important in order to take into account of theeffect of the heterogeneities present in the amorphous silica surface on thewetting of the system. To achieve a suitable force field for the silica-waterinteraction we modify the parameter Cab for the interaction between siliconand water oxygen by fitting the WCA measured in the simulations to theexperimental value of 19.95◦ ± 3.3◦ obtained by Thamdrup et al. [184]. TheWCA measurements are carried out following the procedure employed byWerder et al. [206] but performing a local fit of the circular cross sectionsimilar to Ingebrigtsen and Toxvaerd [102]. We vary systematically the valueof Cab (silicon-water oxygen) with increments of 0.005 kJmol−1nm−2 from0 kJmol−1nm−2 which reproduces a WCA of 97◦ until 0.03 kJmol−1nm−2

which reproduces complete wetting. We observe a linear decrease in thewater contact angle as higher values of the Cab (silicon-water oxygen) areimplemented. The linear behaviour can be evidenced in Figure 6.3, wherethe green line is a linear fit to the date points. Subsequently, we perform alinear interpolation of the data points to match the experimental WCA value.Using the final fitted parameter we reproduce a WCA of 19.8◦ at vacuum asillustrated in Figure 6.2 which is in good agreement with the experimentalmeasurements by Thamdrup et al. [184]. The potential parameters obtainedare specified in Table 6.3. We notice that we did not include in our potentialthe Van der Waals interaction between silica oxygen and water oxygen dueto we did not distinguish between silanol and siloxane oxygen therefore weattempt to calibrate our potential via the silicon-water oxygen interaction.To quantify the strength of the water-silica interaction, we simulate a watermolecule on a silica slab in order to measure the binding energy of a singlemolecule of water close to a amorphous silica surface. We measure a meanbinding energy value of 45 kJmol−1 which is in reasonable agreement withthe results published by Du et al. [58] which vary between 48 kJmol−1 to98 kJmol−1 for a water molecule on H-terminated and pure silica clusters.

67

0

20

40

60

80

100

120

0 0.005 0.01 0.015 0.02 0.025 0.03

WC

A (

degr

ees)

cab (kJ/molnm6)

Figure 6.2: Contact angle of a water nanodroplet (WCA)on a silica substractas a function of the Buckingham parameter Cab: red (+) is the WCA mea-sured at vacuum. The blue (×) is the value fitted to the experimental WCA.The green line is a linear fit.

Parameter Value

αSi−Ow 1.013× 105 kJmol−1

ρSi−Ow 25.00 nm

CSi−Ow 2.360× 10−2 kJmol−1nm6

αOsi−Hw 6.83× 103 kJmol−1

ρOsi−Hw 32.66 nm

COsi−Hw 0.00 kJmol−1nm6

Table 6.3: Buckingham parameters for the silica-water interaction potential.

68

Parameters Value

ǫN−N 0.30264 kJmol−1

σN−N 0.332 nm

ǫO−O 0.43234 kJmol−1

σO−O 0.299 nm

ǫN−O 0.3617 kJmol−1

σN−O 0.3155 nm

ǫN−Si 0.7963 kJmol−1

σN−Si 0.325 nm

ǫN−Osi 0.3075 kJmol−1

σN−Osi 0.293 nm

ǫO−si 0.650 kJmol−1

σO−si 0.318 nm

ǫO−Osi 0.251 kJmol−1

σO−Osi 0.290 nm

Table 6.4: Lennard-Jones parameters for air interactions with silica.

6.2.4 Silica-air interaction potential

We describe the air by modelling the molecular nitrogen and molecular oxy-gen as two Lennard-Jones sites connected with a rigid bond. The interactionsbetween silica and the sites of the molecular nitrogen and molecular oxygenare described employing a LJ potential, using Lorentz-Berthelot mixing ruleswith values obtained from the Universal Force Field (UFF) [155] and fromGuissani et al. [82]. The parameters are shown in Table 6.4. To examinethe reliability of the potential we perform MD simulations of a silica slabsurrounded by an air atmosphere at different air pressures. We equilibratea silica slab as described in subsection 2.1. Subsequently, we perform sim-ulations of the equilibrated silica slab surrounded by air at 100 and 200 barfor more than 4 ns. From the simulations we extract the density profiles ofnitrogen and oxygen. We observe a air interfacial layer with a thickness of ca1 nm as shown in Figure 6.3. In this zone the air density is higher than theair density in the bulk zone. Furthermore, we measure the binding energyof single molecules of N2 and O2 on silica. We obtain a binding energy of22 kJmol−1 for the N2 molecule and of 18 kJmol−1 for the O2 molecule. Sub-sequently, we perform simulations of a nitrogen molecule described using the“pea model” developed by Coasne et al. [43] on a silica slab. We measureda binding energy of 13.5 kJmol−1 which is in the same order of magnitude ofthe values we measure using our molecular nitrogen model.

69

0

50

100

150

200

250

300

0 1 2 3 4 5 6

dens

ity (

kg/m

3 )

distance (nm)

0

50

100

150

200

250

300

0 1 2 3 4 5 6

dens

ity (

kg/m

3 )

distance (nm)

Figure 6.3: Density profile of air at 300 K and 100 bar: density profile ofnitrogen and oxygen in contact with a silica substract. The red line is theN2 density profile and the green line is the O2 density profile.

70

Parameter Value

ǫN−Ow 0.4436 kJmol−1

σN−Ow 0.3243 nm

ǫO−Ow 0.5302 kJmol−1

σO−Ow 0.3078 nm

Table 6.5: Lennard-Jones parameters for water-air interaction.

6.2.5 Water-air interaction potential

The interaction between water and air is simulated employing a Lennard-Jones 6-12 potential (6.2), which is initially parametrized using Lorentz-Berthelot mixing rules. The values for the mixing rules are obtained fromJiang et al. [109] ǫN−N = 0.3026 kJmol−1, σN−N = 0.3320 nm, ǫO−O =0.4320 kJmol−1 and σO−O= 0.2990 nm; and from Berendsen et al. [24] ǫOw−Ow

= 0.6502 kJmol−1 and σOw−Ow= 0.316 nm. The potential parameters for thewater-air cross interactions are listed in Table 6.5. To obtain a suitable po-tential for the interaction between water and air we calibrate the LJ potentialparameters ǫN−Ow and ǫO−Ow using as a criterion the gas solubility in liquidwater. We perform simulations of a water slab including 2800 molecules im-mersed in an atmosphere of N2 and O2 at different pressures. We connectthe system to a Berendsen thermostat for 0.4 ns at 300K, then we disconnectthe thermostat and perform simulations for more than 30 ns until the waterslab is saturated with air. Subsequently, we measure the density of nitro-gen and oxygen dissolved inside water and compute the solubility of the twospecies. We compare the gas solubility values extracted from the simulationsagainst the experimental values [209, 16] listed in Table 6.6. We calibratethe Lennard-Jonnes parameters by systematically varying the ǫN−Ow andǫO−Ow values using as criteria the corresponding gas solubility values ex-tracted from the simulations. We interpolate the ǫN−Ow and ǫO−Ow to matchthe experimental values of the solubility [209, 16] as shown in Figure 6.5.The interpolated parameters are listed in Table 6.7. We perform simulationsusing the calibrated potential parameters and measure the gas solubility. Weobserve a homogeneous solubility of nitrogen and oxygen throughout the wa-ter slab as shown in Figure 6.4 The gas solubility values extracted from thesimulations are listed in Table 6.8. The solubility values are in reasonableagreement with the experimental measurements presented in Table 6.7.

71

0

200

400

600

800

1000

0 2 4 6 8 10 12

dens

ity (

kg/m

3 )

distance (nm)

0

200

400

600

800

1000

0 2 4 6 8 10 12

dens

ity (

kg/m

3 )

distance (nm) (a)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0

dens

ity (

kg/m

3 )

distance (nm) (b)

Figure 6.4: Density of nitrogen and oxygen at 300K and 200 bar in liquidwater as a function of the distance to the center of mass of the water slab.(a) Density profiles of water, nitrogen and oxygen. The blue line is the waterdensity profile, the red line is the N2 density profile and the green line is theO2 density profile. (b) Details of the density profiles of nitrogen and oxygendissolved inside liquid water. The red line is the N2 density profile and thegreen line is the O2 density profile.

72

0

5

10

15

20

25

0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80

solu

bilit

y (c

m3 g

as/g

. wat

er)

ε (kJ/mol)

Figure 6.5: The oxygen and nitrogen solubilities at 300 K and 200 bar asfunction of the Lennard-Jones parameters ǫN−Ow and ǫO−Ow. The greencircles (◦) correspond to ǫO−Ow values and the purple circle (◦) is the ex-perimental solubility value for oxygen. The red crosses (×) correspond toǫN−Ow values and the blue cross (×) is the experimental solubility value fornitrogen.

Specie Pressure Solubility

N2 100 bar 1.265 cm3

gas/gH2O

O2 100 bar 2.417 cm3

gas/gH2O

N2 200 bar 2.257 cm3

gas/gH2O

O2 200 bar 4.621 cm3

gas/gH2O

Table 6.6: Experimental values of the solubilities of nitrogen and oxygen inliquid water. The solubility values included the list have been interpolatedat 300K.

73

Parameter Value

ǫN−Ow 0.5458 kJmol−1

σN−Ow 0.3243 nm

ǫO−Ow 0.5884 kJmol−1

σO−Ow 0.3078 nm

Table 6.7: Calibrated Lennard-Jones parameters for the water-air interac-tion.

Specie Pressure Solubility

N2 100 bar 0.7± 0.3 cm3

gas/gH2O

O2 100 bar 2.7± 0.8 cm3

gas/gH2O

N2 200 bar 2.1± 0.6 cm3

gas/gH2O

O2 200 bar 3.5± 1.1 cm3

gas/gH2O

Table 6.8: Nitrogen and oxygen solubilities in liquid water measured fromthe simulations

6.3 Results and discussion

6.3.1 WCA of water nanodroplets in air at different

pressures

Using the parametrized and calibrated potentials we perform simulations ofa water nanodroplet of 18000 molecules mounted on a silica slab of 37.92 nm× 37.92 nm × 2.45 nm immersed in air (N2 and O2) at air pressures of 50, 100and 150. To equilibrate the silica and the water we follow the methodologyas described in Subsection 2.3 while the air is immobilized. As silica-waterequilibration is reached we release the air molecules to interact with the restof the species until the energy equilibration of the system is reached. Afterthe silica equilibration, we use an integrating time step of 2 fs. Subsequently,we measure the WCA following the methodology described in subsection2.3. We measure the WCA values as shown in the Table 6.9. From themeasured WCA values we observe that the static WCA of the nanodropletis not changing significantly as higher air pressures are imposed.

74

Pressure Water contact angle

vacuum 20◦

50 bar 20◦

100 bar 17◦

150 bar 21◦

Table 6.9: Water contact angle values measured at different pressures.

6.3.2 High density air layers at the silica-water inter-

face

To study the equilibrium properties of silica-water-air systems, we performsimulations using the parametrized force field. The system consists of a waterslab (2800 molecules) on a silica substrate of 3.7 nm × 3.7 nm times 2.45 nmsurrounded by an atmosphere of air at pressures of 50, 100, 200 and 300 bar.In order to equilibrate the system we follow the methodology as describedin subsection 2.3 and 3.1. After the water is saturated with air, which takesmore than 30 ns, we compute the density of water, nitrogen and oxygen as afunction of the distance to the silica slab. Moreover, we compute the solubil-ity of nitrogen and oxygen dissolved inside the water. Density and solubilityprofiles are shown in Figure 6.6 and Figure6.7. From our simulations we infera zone or layer of high concentration of gas adjacent to the silica surface whichhas a thickness of ca 1 nm which can be inferred from Figures 6.6 and 6.7.Moreover, we observe that its thickness and density magnitude seem inde-pendent of the air pressure imposed to the system. Since this interfacial zonewith high gas concentration is present in all of our measurements we believethat it could be ever present in silica-water-air systems. Indeed, this layercould be of significant importance in dewetting, nucleation and stability ofsurfacial nanobubbles, and capillary filling of nanochannels. We measure thebinding energy of silica-water at vacuum and in presence of air at very highpressures as illustrated in Table 6.10. We find that the values of the bindingenergy do not show a significant variation due to the presence of air in water.However, the interfacial layer with high concentration of gas may influencethe dynamics of the interface e.g., through a modified viscosity at the inter-face of silica and water. Recent experiments of capillary filling process ofsilica nanochannels have shown that the liquid is observed to fill at a slowerrate than expected [145, 182, 147, 85]. Various proposals have been made toexplain the observed reduction in the capillary filling rate of nanochannels

75

0

200

400

600

800

1000

1200

0 5 10 15 20 25

dens

ity (

kg/m

3 )

distance (nm) (a)

0

5

10

15

20

25

0 2 4 6 8 10

dens

ity (

kg/m

3 )

distance (nm) (b)

Figure 6.6: Water slab on a silica substrate surrounded by air. Densityprofiles of water and air at 300K and 200 bar. (a) Density profiles for thebulk zone of the system. (b) Solubility of air inside the water slab. The redline is the N2 density, the green line is the O2 density and the blue one isthe water density. The silica surface is at 2.45 nm, it includes geometricalheterogeneities.

76

0

200

400

600

800

1000

1200

0 5 10 15 20 25

dens

ity (

kg/m

3 )

distance (nm) (a)

0

5

10

15

20

25

0 2 4 6 8 10

dens

ity (

kg/m

3 )

distance (nm) (b)

Figure 6.7: Water slab on a silica substrate surrounded by air. Densityprofiles of water and air at 300K and 300 bar. (a) Density profiles for thebulk zone of the system. (b) Solubility of air inside the water slab. The redline is the N2 density, the green line is the O2 density, and the blue one isthe water density. The silica surface is at 2.45 nm, it includes geometricalheterogeneities.

77

Pressure Binding energy

vacuum 209 kJmol−1nm−2

50 bar 226 kJmol−1nm−2

100 bar 212 kJmol−1nm−2

200 bar 222 kJmol−1nm−2

300 bar 212 kJmol−1nm−2

Table 6.10: Silica-water binding energy at different air pressures.

nevertheless the explanation is still under debate [145, 194, 176, 182, 139, 85].Since the interactions between the molecules present in the interface of thesolid surface play a significant role on nanoscale capillary processes [148, 51],we believe the presence of a gas layer at the silica-water interface, as inferredin our simulations, can influence the dynamic wetting [141] and thus affect-ing the capillary filling process in a silica nanochannel. Moreover, we noticethat the presence of an air layer in the silica-water interface could influencethe friction [141] of the system by modifying the viscosity of the interface.Additionally, we hypothetize that the high density air layer can be also theorigin of nanobubbles as proposed from an experimental work by Zhang etal. [219, 220]. Although, the long life time of interfacial nanobubbles con-tradicts the classical theory of bubble stability because their large Laplacepressure would cause a rapid diffusive out-flux of gas [65, 129]. Nevertheless,in recent theoretical and experimental studies some mechanisms explainingtheir stability have been already proposed [72, 138, 220, 219, 34, 55, 59].

6.3.3 Study of capillary filling of water in silica nanochan-

nels

Today, the fundamental relationship between the capillarity and the surfacetension is solidly established, the mechanism of the capillary rise is, in fact,closely related to that of spreading of droplets on a solid substrate. It canbe understood as a spreading in a vertical direction [35]. The surface ten-sion of a liquid is one of the more perceptible manifestations of the forcesthat act between molecules, and the explanation of this phenomena from amolecular approach go back to the eighteenth century. Currently, capillar-ity is an classical topic in physical chemistry nevertheless, on the nanoscale,this topic has an active present as it moves into areas the great interest asnanofluidics. When a wetting liquid is brought into contact with a solid

78

Figure 6.8: Snapshot of a simualtion of the capillary filling process of ananochannel. A liquid consisting of 145000 atoms is filling a hydrophilicchannel driven by capillary force [51].

79

surface in a sufficiently narrow channel, a curved liquid-air or liquid-vaporsurface, the meniscus, is developed due to the surface tensions on the in-terface solid-liquid-gas[35]. The Young-Laplace equation 6.4 describes thecapillary pressure difference across a curved interface, linking the contactangle between the solid-liquid and the liquid-gas interface shape observed atthe contact line of the three phases as illustrated in Figure 6.9.

∆P =2γ cosΘ

R(6.4)

, where ∆ P is the pressure difference across the fluid-gas interface, γ isthe surface tension, Θ is the contact angle and R is the meniscus curvatureradius. The rate of penetration of a liquid into the channel structure isusually determinated using the Washburn equation 6.5, which relates therate of capillary penetration of a liquid to the channel radius (as illustratedin Figure 6.10), the surface tension and viscosity of the liquid and the contactangle between the liquid and the channel surface.

x =

(√

γR cosΘ

2η

)

√t (6.5)

, where x is the distance of penetration of liquid at the time t, γ is the surfacetension, Θ is the contact angle and R is the tube radius. The term into thebrackets is the Washburn coefficient. The contact angle is well determined instatic conditions however for the dynamical state of a contact line as presentin the spreading of droplets or in the advancing meniscus of a capillary fillingprocess, it depends in a complicated way on the particular dynamic state ofthe system [28, 132, 95].

In capillary filling of channels with nanoscale dimensions the effect of thedeviations from the static contact angle of the meniscus can not be neglectedas noted by Blake at al. [28] and Martic et al [132]. Moreover, in capillaryflow inside nanochannels, capillary can not be separate from the characteristicsizes of the system therefore heterogeneities in the structure of the interfacialzone such as gas layers, nanobubbles or dewetting, have to be taken intoaccount. In recent experimental works of water capillary flow in nanometersize channels, Tas et al. [182], Persson et al. [147], Haneveld et al. [85] and Ohet al. [145] find quantitative deviations from the theoretical predictions usingthe classical Washburn equation. Persson, Oh and Haneveld measured forsilica channels with heights below 100 nm a smaller than expected Washburn

80

0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8 1

pres

sure

(ba

r)

radius (µm)

Figure 6.9: Sketch of the Young-Laplace equation. The difference of pressureas a function of the curvature radius of the interface water-air for a contactangle of 20 ◦ and a surface tension of 73mJ/m2.

81

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5

dist

ance

(m

m)

time (s)

Figure 6.10: Sketch of the Washburn equation modified to model the capillaryfilling of a rectangular cross section channel with height of 10 nm and width≫ h. The propagated length of the water meniscus is plotted as a functionof time. The water contact angle is 20 ◦, the surface tension is 73mJ/m2 andthe viscosity is 0.89 Pa s.

82

(1)

(2)

(3)

(4)

Figure 6.11: Sequence of snapshots from a MD simulation of the capillaryfilling process of a water slab in a silica channel at vacuum with height of10 nm.

83

Pressure velocity

vacuum 15 nmns−1

10 bar 10 nmns−1

20 bar 0 nmns−1

75 bar 0 nmns−1

150 bar 0 nmns−1

Table 6.11: Mean axial velocities of the advancing meniscus of water in asilica nanochannel of 10 nm height at different air pressures.

coefficient. The observed deviation of Washburn coefficient is higher fordecreasing channel heights. Various suggestions have been argued to explainthe observed reduction in the capillary filling rate of silica channels howeverthe explanation is still under debate [176, 182, 139, 85]. As we noticed in thisthesis we infer from our simulations of the silica-water-air interface a zone,placed very close to the silica surface, where the concentration of nitrogenand oxygen are significantly higher than the concentration of these gases inthe bulk water. We think that this layer could affect the velocity of theadvancing meniscus via a modified viscosity. Additionally, we believe thatthe high density air layer can be also the origin of nanobubbles as it wasproposed from an experimental work by Zhang et al. [219, 220] thereforenucleaded nanobubbles could influence the slip boundary condition at theinterface [205]. Although, to date the evidence for the existence of interfacialnanobubbles is still contradictory nevertheless some mechanisms explainingtheir stability have been already proposed [72, 138, 220, 219, 34, 55, 59]. Ourhypothesis is consistent with the experimental observations of capillary fillingof nanochannels [147, 85, 184, 145] since the deviations in the Washburncoefficient are higher as the heights of the channel are smaller, it meansthe impact of the presence of the gas layer on the flow rate increase as therelation between the channel height and the layer thickness is closer. Toconfirm the presence of air at high pressure affects the capillary filling speedin channels, we perform simulations of a water slab filling an amorphoussilica channel of 10 nm height at different high air pressures (0, 10, 20, 75and 150 bar) as shown in Figure 6.11, from the simulations we extract themean axial velocity of the advancing water slab (Figure 6.12. We notice thatthe water meniscus is not advancing as air pressures higher than 20 bar areimposed to the system. Moreover, the mean axial velocity of the water slabdecreases as higher air pressures are imposed as shown in Table 6.11. In fact,

84

0

5

10

15

20

25

30

35

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

velo

city

(nm

/ns)

time (ns) (a)

-5

0

5

10

15

20

25

30

35

40

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

velo

city

(nm

/ns)

time (ns) (b)

Figure 6.12: Capillary filling speed of water in a silica nanochannel of 10 nmheight. (a) Mean axial velocity of the advancing meniscus of water at vacuumas a function of the time. (b) Mean axial velocity of the advancing meniscusat 10 bar as a function of the time.

85

Pressure Water contact angle

vacuum 45◦

10 bar 70◦

20 bar 80◦

75 bar 90◦

150 bar 90◦

Table 6.12: Water contact angle of the advancing meniscus at different airpressures.

we observe a rise in the dynamic contact angle of the advancing meniscusas higher pressures are imposed as shown in Table 6.12. In our previousstudy of the effect of air on the static WCA of a water droplet on a silicasubstrate at different pressures, no changes in the static WCA are observedas we vary the air pressure around the droplet. Therefore, we infer that theair confined inside the channel is affecting only the dynamic contact angleof the water meniscus which could conduct to the observed decrease in themean axial velocity as it was inferred by Han et al. [84] for capillary fillingof nanochannels with ethanol, isopropanol and water.

6.4 Conclusions

We have parametrized and calibrated a MD force field which is suitable tosimulate nanofluidic phenomena in silica-water-air systems. We perform sim-ulations of water nanodroplets on silica surfaces at different air pressures inorder to study the role of the air on the WCA. Moreover, we conduct MDsimulations of bulk water on a silica slabs to investigate the silica-water inter-face in presence of air at different pressures. Adjacent to the immersed silicasurface we observe a zone of ca 1 nm of thickness where a higher concentrationof gas is measured. We hypothesize that this gas layer could be part of theexplanation of unusual phenomena related to capillary filling of nanochannelsobserved experimentally. Furthermore, we believe that this layer with highdensity of gas could promote the nucleation of nanobubbles and influencingits stability on a silica substrate. Subsequently, we perform MD simulationsof the water capillary filling of a silica nanochannel of 10 nmheight. We be-lieve that the presence of air in the system could be the origin of two differenteffects which simultaneously can influence the capillary filling rate. The airdissolved in the liquid produces a high density air layer very close to the silica

86

substrate as shown in Figure 6.7. This air layer affect the filling rate via a de-creased viscosity in the interface between the water and the silica substrate.In addition, the air in the channel could affect the capillary force by a modi-fication of the dynamic contact angle of the advancing meniscus as observedin our simulations (Table 6.10). We attempted to perform MD simulationsto study directly the effect of interface nanobubbles on the capillary fillingrate however due to the required size of the system and the very long simu-lation time needed to evaluate the stability of nanobubbles, we were not ableto accomplish this target with our computational resources. Nevertheless,we believe that our results provide useful contributions to the study of thecapillary phenomena at the nanoscale and encourage future computationalstudies of this type of systems.

87

Chapter 7

Summary and Future Work

In this thesis molecular dynamics simulations have been performed to studydifferent types of basic nanodevices. A study of the suitability to use sin-gle wall carbon nanotubes as liquid conduits is accomplished including andetailed investigation of the possibility of the exploitation of thermal gra-dients as driving mechanism to move liquids on the nanoscale. Moreover,molecular dynamics simulations are used to analyze thermophoretic linearmolecular motors fabricated of coaxial carbon nanotubes. Finally, new forcefields have been calibrated to model silica, water and air in order to simulatenanofluidic systems. Using the calibrated force fields, simulations of waternanodroplets mounted on silica substrates and of water capillary filling ofsilica nanochannels are performed to investigate the influence of air on thecapillary filling process in nanoscale channels. As we noticed in the conclu-sions of the chapter 6, we believe that more powerful computational resourcesare needed to study directly the influence nanobubbles on the capillary fillingprocess of nanochannels. We attempt as a future target the development ofa more complex and realistic air model taking into account the quadrupoleeffect [43] of the electric charges at the nitrogen and oxygen atoms. Never-theless, it is not clear if the inclusion of that quadrupole effect could provideimproved results in the analysis of the capillary filling process in presence ofair. With this thesis, I wish contribute to the advance of the understandingof some basic concepts necessary for the development of fluidic nanodevicesand to encourage the future exploitation of molecular dynamics simulationsas a standard analytical tool to study nanofluidic phenomena related to thedevelopment of nanodevices.

88

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