• Aucun résultat trouvé

Atomistic engineering of fluid Structure at the fluid-solid interface

N/A
N/A
Protected

Academic year: 2021

Partager "Atomistic engineering of fluid Structure at the fluid-solid interface"

Copied!
141
0
0

Texte intégral

(1)

Atomistic Engineering of Fluid Structure at the

Fluid-Solid Interface

by

Gerald

J.

Wang

B.S. Mechanical Engineering, Math & Physics, Yale University (2013)

S.M. Mechanical Engineering, MIT (2015)

Submitted to the Department of Mechanical Engineering

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Mechanical Engineering and Computation

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

February 2019

Gerald

J.

Wang, MMXIX. All rights reserved.

The author hereby grants to MIT permission to reproduce and to

distribute publicly paper and electronic copies of this thesis

document in whole or in part in any medium now known or

hereafter created.

Signature

redacted-Autfhor

Certified by...

Accepted by ..

Accepted by ..

MASSACHUSTS 1NSjn1JTOf TECHNOLOGY

FEB

'5

2019

LIBRARIES

ARCHIVES

...

Department

ical ngineering

Signature redacted

2019

... ...

Nicdlas G. Hadjiconstantinou

Profeso

ec nical Engineering

Signature

redacted

esi

pervisor

... ...o. ...

NIcolas G. Hadjiconstantinou

Chairman, Cojmpittegon Graduate Students

...

Signature redacted

(

Youssef Marzouk

Co-Director, Computational Science and Engineering

I

(2)
(3)

Atomistic Engineering

of Fluid

SfrUrtuire at

the

Fluid-Solid Interface

by

Gerald

J.

Wang

Submitted to the Department of Mechanical Engineering on January 15, 2019, in partial fulfillment of

the requirements for the degree of

Doctor of Philosophy in Mechanical Engineering and Computation

Abstract

Under extreme confinement, fluids exhibit a number of remarkable effects that cannot be predicted using macroscopic fluid mechanics. These phenomena are especially pronounced when the confining length scale is comparable to the fluid's internal (molecular) length scale. Elucidating the physical principles governing nanoconfined fluids is critical for many pursuits in nanoscale engineering. In this thesis, we present several theoretical and computational results on the structure and transport properties of nanoconfined fluids.

We begin by discussing the phenomenon of fluid layering at a solid interface. Using molecular-mechanics principles and molecular-dynamics (MD) simulations, we develop several models to characterize density inhomogeneities in the interfa-cial region. Along the way, we introduce a non-dimensional number that predicts the extent of fluid layering by comparing the effects of fluid-solid interaction to thermal energy. We also present evidence for a universal scaling relation that re-lates the density enhancement of layered fluid to the non-dimensional temperature, valid for dense-fluid systems.

We then apply these models of fluid layering to the problem of anomalous fluid diffusion under nanoconfinement. We show that anomalous diffusion is controlled by the degree of interfacial fluid layering; in particular, layered fluid exhibits restricted diffusive dynamics, an effect whose origins can be traced to the (quasi-) two dimensionality and density enhancement of the fluid layer. We construct models for the restricted diffusivity of interfacial fluid, which enables accurate prediction of the overall diffusivity anomaly as a function of confinement length scale.

Finally, we use these earlier developments to tackle the notorious problem of dense fluid slip at a solid interface. We propose a molecular-kinetic theory that formulates slip as a series of thermally activated hops performed by interfacial fluid

(4)

molecules, under the influence of the bulk fluid shear stress, within the corrugated energy landscape generated by the solid. This theory linearizes to the Navier slip condition in the limit of low shear rate, captures the central features of existing models, and demonstrates excellent agreement with MD simulation as well as experiments.

Thesis Supervisor: Nicolas G. Hadjiconstantinou Title: Professor of Mechanical Engineering

(5)

Acknowledgments

Most answers in research beget at least two new questions; in this sense, no section of this thesis is particularly complete. But there is no section that is more systemically incomplete than this one, because I could not possibly (in finite time and to sufficient extent) list all of the positive forces that have empowered me to pursue this work, enriched my thought process, and brought joy to the last five years. Nevertheless, I must try. What follows is, to me, the most cherished and celebrated of the \begin{enumerate}'s in this document. I would like to thank...

1. Prof. Nicolas Hadjiconstantinou. He has been a near-infinite reservoir of

knowledge, inspiration, and good humor. I have not yet come close to equilibrating with this reservoir; but, as I have learned often under his guidance, some of the best things in life are not in equilibrium.

2. Profs. Rohit Karnik, Petros Koumoutsakos, and Ju Li. Although their last initials are (alphabetically speaking) confined to a surprisingly small do-main, their research mentorship has been anything but. I feel exceptionally lucky to have received their ideas, advice, and encouragement.

3. The Department of Energy Computational Science Graduate Fellowship,

the Krell Institute, and all of my fellow fellows. I am very grateful to this community for its generous camaraderie and support.

4. Rachel Kurchin. From lentils to long runs, semiconductors to support-iveness, comedy to complementarity, she makes me a better person in innumerable ways (alliterative and otherwise).

(6)

5. Nickolas Demas, Mimi Nakajima, and Nisha Chandramoorthy. I have

almost surely struck upon a set of measure zero in finding such terrific co-conspirators, confidantes, and companions.

6. Michael Boutilier, Sam Chevalier, Boyu Fan, Margaux Filippi, Mojtaba

Forghani, Ashkan Hosseinloo, Hussain Karimi, Colin Landon, Hung Nguyen, J.-P. Peraud, Rohit Supekar, Mathew Swisher, and Essie Yu. Their incom-parable company has made the Nexus simultaneously a home (e.g. of loved ones) and a battlefield (e.g. of ideas).

7. The Edgerton House community. They have indelibly impressed on me

the joys of living in a parallelogram, which I shall never forget.

8. Emilie Heilig, Leslie Regan, and the entire MechE Grad Office. Thanks to

their effortless enthusiasm, it can be easy to forget that they all routinely

solve problems with > 6N degrees of freedom (N - 103).

9. My colleagues in the group of Prof. James Swan at MIT and in CEE at

Carnegie Mellon University. In no time at all, you've made me feel very much at home. I eagerly anticipate the excitement and adventures that correspond to the finite-time limit.

10. Henry Wilkin, Tong Zhan, Rick Herron, and the Shedd and Kurchin

fami-lies. I am thankful for their friendship, which is one of the primary reasons

I have ever wandered out of Central and Kendall Squares.

11. Last, and most of all, my wonderful family - my mother Hong, my father Jie, my sister Jennie, the cat Simba, and many others. I owe everything that I am to them (minus the cat).

(7)

Contents

1 Introduction 21

1.1 Big Surprises Come in (Very) Small Packages . . . . 21

1.2 Challenges in Nanoscale Fluid Mechanics . . . . 23

1.2.1 Fluid Layering on a Solid Interface . . . . 23

1.2.2 Fluid Diffusion under Nanoconfinement . . . . 27

1.2.3 Fluid Slip at a Solid Interface . . . . 28

1.3 Thesis Overview . . . . 29

2 Fluid Structure at a Solid Interface 31 2.1 Fluid Layering in Semi-Infinite Systems . . . . 33

2.1.1 The Wall Number and its Thermodynamic Interpretation . 34 2.1.2 The Degree of Layering and the Layering Regime . . . . . 35

2.1.3 Anatomy of the First Layer: Length Scales of Layering and the Density Enhancement . . . . 36

2.1.4 Molecular Mechanics of the First Layer . . . . 42

2.2 Extension to Finite Systems . . . . 58

2.3 Generalizations to Other Fluid-Solid Systems . . . . 59

2.3.1 Effect of Boundary Structure . . . . 61

2.3.2 Effect of Boundary Rigidity . . . . 62

2.3.3 Effect of LJ Parameters . . . . 63 7

(8)

2.4 Summary and Discussion ... 63

3 Fluid Diffusion under Nanoconfinement 67 3.1 Background and Objective . . . . 69

3.2 Preliminary Observations . . . . 72

3.2.1 Spatially Resolved Diffusivity . . . . 72

3.2.2 Particle Residence Time within the First Layer . . . . 72

3.2.3 Characteristic Time Scales . . . . 75

3.3 Layering and Anomalous Diffusivity . . . . 77

3.3.1 The First Layer as a "Dimensionally Restricted" Fluid . . . 79

3.3.2 Anomalous Diffusivity Magnitude . . . . 81

3.3.3 Confinement-Induced Anomalous Diffusivity . . . . 83

3.4 Two Detailed Approaches for the Anomalous Diffusivity in the First Layer . . . . 86

3.4.1 An Approach Based on the Statistical Physics of Hard Spheres . . . . 86

3.4.2 An Approach Based on Excess Entropy . . . . 90

3.5 Conclusions . . . . 97

4 Fluid Slip at Solid Interfaces 99 4.1 Background and Objective . . . 100

4.2 Theory . . . 103

4.2.1 Potential Energy Landscape Near a Solid Surface . . . 103

4.2.2 Molecular-Kinetic Model for Slip . . . 103

4.3 Molecular Simulations . . . 105

4.3.1 Varying Shear Rate . . . 106 4.3.2 Varying Temperature and Fluid-Solid Interaction Strength 110 4.4 D iscussion . . . . .113

(9)

4.5 Conclusions . . . 114

5 Summary and Future Directions 117 5.1 Summary . . . 117

5.2 Future Directions . . . 119

A Molecular-Dynamics Simulations 123 A. 1 Fundamentals of Molecular-Dynamics Simulations . . . 123

A.1.1 The Basic MD Algorithm . . . 124

A.1.2 Interaction Potentials . . . 126

A.2 Systems for Chapter 2 . . . 127

A.3 Systems for Chapter 3 . . . 127

A.4 Systems for Chapter 4 . . . 128

(10)
(11)

List of Figures

1-1 Visualization of equilibrium Lennard-Jones fluid structure within

a CNT (R =24 A) obtained from MD simulation. Blue beads denote fluid atoms and gray edges denote bonds within the CNT.

Of particular note is the arrangement of fluid in concentric rings

near the CNT wall. . . . . 24 1-2 Representative radial density function for fluid confined within

a CNT at 300 K, including a sharp density peak at a distance of approximately 3 A from the solid. . . . . 25

2-1 Schematic illustrating the definition of the degree of layering E, which is calculated as the ratio of the magnitude of the first peak on the fluid density profile (shown in green) to the magnitude of the first minimum between the first two peaks (shown in orange). 36

2-2 Relation between degree of layering and Wall number, obtained from 33 MD simulations of LJ fluid confined within a graphene nano-slit with 0.4 Pave 1.2 and 0.6 < T 20. . . . . 37

2-3 Fluid layering near the graphene surface, obtained from MD

sim-ulation at T = 6.4 and Pave = 1.2 (Wa = 0.6). The minimum

separation Zmin is schematically illustrated in orange. . . . . 38 11

(12)

2-4 Four smoothed fluid spatial density profiles, corresponding to low density (Pave = 0.5) and high density (Pave = 1.0) as well as

low temperature (T = 3.2) and high temperature (T = 6.4). In all cases, density variations are low in amplitude at a distance of 5a

from the wall. The first-layer width hFL is schematically illustrated in green. Note that hFL is a function of density, and also a weak

function of temperature in the low-density case. The minimum separation Zmin given in Equation (2.8) is schematically illustrated in orange. Note that this quantity does not discernibly depend on density or temperature. . . . . 39

2-5 MD results (0.6 T 20) for the stand-off distance Zmin are shown as a function of density, along with the constant value predicted by mean-field theory in Equation (2.8). Note that MD measurements

of Zmin are nearly independent of the temperature and fluid density. 40

2-6 Density enhancement as a function of temperature for 0.7 Pave <

1.2 (inset shows the density enhancement for three additional low densities, 0.4 Pave 0.6). Note that the density enhancement

decreases with increasing temperature for all densities. Moreover, for Pave 0.7, the enhancement can be accurately modeled using

7FL from Equation (2.28) (represented by dashed curve) as well as

using s-2 from Equation (2.36) (represented by diamonds). . . . . 43

2-7 MD results for the first-layer areal densities EFL as a function of

temperature T. Equation (2.28), represented by dashed curves, is able to capture these values of EFL accurately (especially for

(13)

2-8 The first-layer width hFT is shown for three different densities and

3.2 T 6.4. Note that hFL is clearly a function Of Pave but only a weak function of temperature . . . . 50 2-9 MD results for the first-layer width hFL are shown as a function of

density, along with a linear fit. The bars indicate the variation of first-layer widths due to temperature in the range 2 T 20. . . 51

2-10 Density enhancement as a function of the average fluid density, for dense fluids (pave > 0.7), based on numerical evaluation of

(2.33) at five different temperatures. Note that there is a gradual

decrease in C towards unity with increasing T, and only small

variation (< 5% for T > 4) of C with Pave. . . . . 54

2-11 Radial distribution functions for fluid atoms within the first layer, for systems with six different densities, obtained from MD simu-lations at T = 3.8. . . . . 56 2-12 MD results and corresponding fits to Equation (2.36) for s-2, as a

function of fluid average density, at three values of temperature. . 57 2-13 The predicted value of C, based on the value of EFL from Equation

(2.28), is shown against results from MD simulations of fluid

con-fined within small systems (L = 15.9 and L = 6.3), with agreement w ithin 12% . . . . . 60

3-1 Fluid density (relative to the channel-averaged density, Pave = 1.0)

as a function of distance from the wall. This representative profile shows key features of fluid layering near the fluid-solid interface at high Wall number (Wa = 1.2). The minimum separation Zmin is marked in orange, the first-layer width hFL is marked in green, and the first-layer wavelength AFL is marked in purple. . . . . 70

(14)

3-2 The characteristic time Tjump taken by fluid molecules to leave the

first layer is shown as a function of the Wall number (simulations performed by varying T at fixed n). The blue color (Wa ;_ 1) indicates the region over which exponential fit is based. For all densities, Tjump is increasing in Wa; in particular, the exponential

fit for layer-forming systems indicates that particle escape from a well-formed fluid layer is consistent with a thermally activated hopping mechanism. . . . . 74

3-3 Mean-squared displacement vs. time for particles in the bulk

region (L = 15) obtained from MD simulation. For At ;> =

O.1L2/(T[ 2

Dbuik), the bulk fluid enters the "2D regime" as it

be-gins to exhibit confinement effects associated with the finite-sized channel. Note that the transition regime associated with CFL (shown in purple) is significantly shorter than that associated with Tjump (shown in yellow), which is in turn much smaller than the "3D regime" (shown in green). As a guide to the eye, and to emphasize that the MD result deviates from the bulk 3D behavior, the bulk 3D line is extended beyond the 3D regime by the green dotted line. . . . . 78

3-4 Mean-squared displacement as a function of time for particles within the bulk (dark green line), for particles within the first layer (orange line), and for the z-component of particles in the first layer (blue line). Note the difference in slopes between At _ Tjump

(the "dimensionally restricted" and transition regimes, shaded in red and yellow, respectively) and At >> Tjump (the "dimensionally unrestricted" regime, shaded in green). . . . . 80

(15)

3-5 Diffusivitv ratio as a function of the Wall number for a variety of densities and channel widths. Note that in the low-Wa regime,

Q ~1 for all densities and channel widths; in the high-Wa regime,

Q

can be considerably less than unity. . . . . 82 3-6 Diffusivity ratio

Q,

as a function of confinement length scale L for

systems in the layering regime, Wa > 1 (Pave = 0.8). The colors

run from white (Wa = 1.0) to blue (Wa = 3.0). The inset shows

the diffusivity anomaly for several systems in the low-Wa regime. The colors run from white (Wa = 1.0) to red (Wa = 0.2). . . . . 85 3-7 Ratio of first-layer diffusivity to bulk diffusivity as a function of

temperature (pave = 0.8), as measured in MD simulations and as

predicted by Equation (3.16). The latter successfully predicts that DFL/Dulk increases with T, eventually approaching a value near

2/3 (the loose upper bound based on dimensional restriction). . . 88

3-8 Diffusivity anomaly as a function of the confinement length scale, using the values of DFL/Dbulk calculated via Equation (3.16) and Equation (3.10). These predicted values are in good agreement with the MD results shown in Figure 3-6 for a variety of Wa > 1.. 89

3-9 Schematic illustration of the method for calculating excess entropy in the first layer. The particles within the first layer (Zmin z

zmin + hFL) are identified and then projected onto the plane of the

solid i.e. the z coordinate is projected out. The two-dimensional pair correlation function is calculated on the projection; this func-tion is then used in Equafunc-tion (3.19) to calculate the excess entropy in the first layer. . . . . 92

(16)

3-10 Scaled diffusivity as a function of the first-layer excess entropy.

The dots indicate measurements obtained from MD simulation via the procedure illustrated in Figure 3-9 (0.8 Pave < 1.0, 0.6 <

T 7.6); the dashed line is an exponential fit to Eq. (3.21). . . . . . 93

3-11 Ratio of first-layer diffusivity to bulk diffusivity as a function of

temperature (0.8 Pave 1.0). The dots indicate measurements

directly from MD simulation via measurements of mean-squared displacement as given in Equation (3.2). The crosses indicate the values of DFL/Dulk that are predicted using Equation (3.22), based on measurements of the excess entropy, using the pair correlation function, in both the bulk and the first-layer regions. The dashed line is an exponential fit, using non-linear least squares, to the results obtained via excess entropy (crosses). . . . . 95

3-12 Diffusivity anomaly as a function of the confinement length scale,

using the values of DFL/Dik calculated via Equation (3.22) and

Equation (3.10). These predicted values are in good agreement with the MD results shown in Figure 3-6 for a variety of Wa > 1. . 96

4-1 Schematic representation of slip in a plane-Couette geometry. The fluid velocity profile in black represents the no-slip condition; the velocity profile in red exhibits an amount of slip shown in purple as us. The slip length L, can be geometrically interpreted as the distance beyond the boundary b that the profile with slip needs to be extended in order to recover the boundary velocity. . . . 101

(17)

4-2 Schematic representation of hopping of fluid molecules within a corrugated potential energy landscape (in 1D). The addition of an external force, indicated in red, tilts the landscape and makes hopping in the forward direction more likely than hopping in the backward direction. . . . 104 4-3 Scaled slip velocity as a function of scaled shear as measured in

MD simulations ([1] in green, [2] in pink, and our work in blue). The scaling predicted by Equation (4.7) is shown as the dashed green line. . . . 108 4-4 Scaled slip velocity as a function of scaled shear for experiments

performed via atomic force microscopy (AFM, [3]), surface-force apparatus (SFA, [4]), and near-field laser velocimetry (NFLV, [5]). The scaling predicted by Equation (4.7) is shown as the dashed green line. . . . 109 4-5 Slip velocity as a function of temperature for high- and

low-density fluids (u = 0.25), where the change in non-dimensional

temperature is effected by changes in the dimensional tempera-ture (E is held constant at 1). For both densities, the results from MD simulation show strong agreement with the dependence in

(4.10). . . . .111

4-6 Slip velocity as a function of fluid-solid interaction strength for high- and low-density fluids (u, = 0.125). For both densities,

the results from MD simulation show strong agreement with the exponential decay predicted by Equation (4.6). . . . 112

(18)
(19)

List of Tables

4.1 Parameters for characteristic length and time scales in (4.6), for each of the densities in Figure 4-3. . . . 107

(20)
(21)

Chapter 1

Introduction

"To get where we want to go, we need to do the small things."

- Joe Maddon

1.1 Big Surprises Come in (Very) Small Packages

Fluids under nanoscale confinement, unlike their macroscale counterparts, ex-hibit a number of remarkable effects that cannot be predicted within a continuum theoretical framework, namely, through the Navier-Stokes equations. Because these very small systems possess surface-area-to-volume ratios that are dramat-ically larger than those that are found in bulk fluids (fluids under macroscale confinement), the physics of nanoconfined fluids is dominated by interfacial phenomena, which can significantly impact both equilibrium and transport properties. Fundamentally, these effects stem from the fact that under such tight confinement, the atomistic features of a fluid play a major role; in particular, in a nanoscale system, the confining length scale is comparable to the fluid's internal length scale.

Elucidating the physical principles governing nanofluidic flows is critical for many pursuits in nanoscale technology. These nanofluidic phenomena af-ford great opportunities to think big by thinking small. For example, in the

(22)

water-energy nexus, nanoscale fluid physics plays a critical role in designing nanoporous desalination devices [6, 7, 8, 9], tailoring thermal transport at a fluid-solid interface [10], and harvesting energy from unconventional sources, such as salinity gradients [11]. In the biomedical domain, nanofluidic flows are relevant for engineering nano-syringes, which could potentially perform drug delivery across cell membranes [12, 13]. Accurate estimation of nanoscale fluid properties may play a key role in assessing the viability of nanoporous rock for hydrocarbon extraction [14] or carbon sequestration [15].

An accurate understanding of nanoscale flow physics is especially impor-tant for modeling anomalous fluid flow rates through carbon nanotubes (CNTs)

[16, 17, 18], which can be several orders of magnitude faster than predicted by classical fluid mechanics [19, 20, 21, 22, 23]. Analytically modeling these

anomalous properties can also be very beneficial from a computational engi-neering point of view, since it allows realistic simulation of nanofluidic systems without resorting to computationally expensive techniques such as coupling to an external fluid bath [24, 25, 26] or liquid-state density-functional theory [27]. From the perspective of efficient and accurate multi-scale simulation, it is highly desirable to incorporate molecular-scale features into macroscopic (continuum) solvers [28].

Despite the immense promise of nanoscale fluid mechanics, there are still numerous gaps in the literature on the physics of dense fluids under nanocon-finement; we discuss several important examples in greater detail in the next section. The fundamental difficulty that hampers the development of theories for dense fluids is that dense fluids lack a small parameter. In contradistinction, dilute fluids (e.g. rarefied gases) have low densities, and can be studied using kinetic theory [29,30]; on the other hand, solids have small atomic displacements from equilibrium, and can be studied using wave theory and collective modes

(23)

[31]. As a consequence, we cannot construct models for dense fluids upoL

fa-miliar theoretical scaffolding (e.g. closed-form equations of state or dispersion relations). To this end, the central goals of this thesis are to identify the struc-tures (in both a conceptual and literal sense) that underpin dense nanofluidic phenomena and explore how these structures relate to nanofluidic transport. As interest in nanoscale engineering continues to grow, there will be an ever in-creasing need for fundamental theories that accurately capture nanoscale effects in dense fluids.

1.2

Challenges in Nanoscale Fluid Mechanics

In this section, we will introduce, motivate, and review the progress on several significant problems in nanoscale fluid mechanics. These problems constitute the core motivation for this thesis, and highlight the intimate connections be-tween nanofluidic structure and transport.

1.2.1

Fluid Layering on a Solid Interface

Over the past five decades, a considerable body of literature has been written on the nanoscale structure of fluids at the fluid-solid interface. In the late 1970s, it was first discovered in molecular simulations that fluid in the vicinity of a solid planar boundary can arrange in a layered structure that runs parallel to the boundary [32, 33]. Numerous molecular-dynamics (MD) studies have also demonstrated fluid layering in other confining geometries, most notably within carbon nanotubes (CNTs) [24, 34, 351. These studies have established that fluids confined within nanoscale domains will form ordered layers near the solid boundary, whereas fluid far from the fluid-solid interface exhibits little ordering

(24)

-, -1

and has properties that resemble bulk fluid [24, 36, 37]. These features can be observed in Figure 1-1, based on an MD simulation that we have performed using LAMMPS [38], which shows the structure of a van der Waals fluid in a

CNT of radius 24

A.

This phenomenon of distinct layers forming at the fluid-solid interface has also been observed in experiments, most notably through ultrafast electron crystallography [39].

Figure 1-1: Visualization of equilibrium Lennard-Jones fluid structure within a CNT (R =24

A)

obtained from MD simulation. Blue beads denote fluid atoms and gray edges denote bonds within the CNT. Of particular note is the arrange-ment of fluid in concentric rings near the CNT wall.

Many computational studies (see, for example, [40, 41]) have also reported the presence of a stand-off distance between the solid boundary and the fluid

(25)

25 -T =_300 K 2015 -E 0

2:

1 5-0 0 50 100 150

Distance from Solid

[0. 1 A]

Figure 1-2: Representative radial density function for fluid confined within a

CNT at 300 K, including a sharp density peak at a distance of approximately 3

0

A from the solid.

layering (this feature can be seen in Figure 1-1). The existence of a stand-off distance has also been demonstrated via experimental investigations of liquid-solid interfaces using scanning tunneling microscopy [42]. In both simulations and experiments, this stand-off distance has been determined empirically to be on the order of one atomic diameter. Moreover, it has been shown in non-equilibrium MD simulations that fluid flow does not affect appreciably affect these stand-off distances as compared to equilibrium MD simulations [43, 44].

The first layer of adsorbed fluid, which is formed just outside this stand-off distance, tends to be of substantially higher density than the bulk density; this phenomenon is illustrated in Figure 1-2, also based on an MD simulation that we

(26)

have performed. The magnitude of this high-density layer has a considerable dependence on temperature, bulk fluid density, and the fluid-solid interaction strength [26, 45, 46, 47]. However, prior to this thesis, no model has been proposed that accurately captures the effects of these parameters on the fluid structure near the solid interface.

Layering can have profound effects on the properties of a fluid under nanocon-finement. Early work focused on the phenomenology of layer formation, in-cluding the structural features of adsorbed layers [48] and their implications for tribological applications [49]. More recently, several groups have reported that when a nanoconfining structure (e.g. a CNT or a graphene nano-slit) is held in equilibrium with a large reservoir of fluid, the density of the fluid confined within the nanostructure is measured to be substantially less than the density of the fluid reservoir [35, 34, 50]. We have previously shown that this density anomaly is the direct result of fluid layering [51]. Layering has also been shown

to be closely related to the solvation pressure in nanoconfined fluids [52, 53]. It is worth emphasizing that understanding fluid layering is a gateway to pre-dicting and controlling many nanofluidic transport phenomena beyond the ones described above. As is well known, fluid layering is at the heart of many elec-trokinetic phenomena. For example, the electric double layer (layering within an ionic fluid near a charged solid interface, plays an important role in the trans-port of ionic solutions through small-scale devices [52]. In the thermal sciences, MD simulations [47] (as well as experiments [54]) have shown that the thermal resistance at the liquid-solid interface depends strongly on the magnitude of fluid layering at the interface. This connection between interfacial fluid struc-ture and thermal resistance was recently extended to liquid-vapor systems [55]. Consequently, it may be possible to indirectly control the transport properties in nanofluidic devices by directly controlling the magnitude of fluid layering at

(27)

the interface (this could be accomplished, for examp, by I i nin C4e fl 1;d-s%1id

interaction strength [26, 45, 46] or strain engineering [56]). Since the first fluid layer can be of considerably higher density than the bulk, it may even be possible that fluid layers near the solid interface can support phonon-like modes, which has profound implications for solid-to-liquid heat transfer in nanoscale devices [46].

In this thesis, we tackle two specific examples of transport properties that are closely related to fluid layering, which we discuss below.

1.2.2

Fluid Diffusion under Nanoconfinement

Under nanoconfinement, the diffusive behavior of fluids may differ significantly from their bulk counterparts. Anomalous diffusive behavior is of great engineer-ing interest in many nanofluidic systems. In general, diffusion plays a dominant role in mass transfer for systems with small pore radii where it may be challeng-ing to impose pressure gradients that are sufficiently large to drive convection. Studying equilibrium diffusion in nanoscale systems also sheds light on trans-port within these systems, by way of the fluctuation-dissipation relations that connect equilibrium and transport quantities. This approach has been fruitfully pursued in the literature, as reviewed in [57].

The phenomenon of anomalous diffusion has been observed in a wide range of fluid simulations, including simulations of hard-disk/sphere fluids [58, 59,

60, 61], Lennard-Jones fluids [62, 63, 64, 58, 65], water [66, 67, 68], oxygen [69],

and a wide variety of alkanes [70]. The observation that confinement can induce anomalous diffusive dynamics is also supported by a wide range of experimental studies [71, 72, 73]. Despite these results, there is a considerable gap in the literature for models that can explain the origins of anomalous diffusive behavior

(28)

using molecular mechanics and knowledge of the layered fluid structure that emerges under confinement.

1.2.3

Fluid Slip at a Solid Interface

Whereas the no-slip boundary condition is commonly used for modeling macro-scopic flows, slip at the fluid-solid interface can play a crucial important role in nanoscale flows; in particular, slip serves to "speed up" nanoscale flows relative to their bulk counterparts [74]. Unsurprisingly, nanoscale fluid slip has received considerable attention (see, for example, [1, 75, 76, 77, 78, 79]). In the case of dilute gases, the functional form of the slip relation as well as slip coefficients can be calculated via asymptotic expansions of the Boltzmann equation [30, 80]. However, in dense liquids such analytical treatments are not possible.

Most research in the dense-fluid arena has thus focused on investigating the properties of the slip length P, which is the distance into the solid that the fluid velocity profile would have to be extrapolated in order to recover the boundary velocity. Of particular note is the work of Thompson and Troian [1], which showed that MD data for the slip length could be described well by an expression

of the form

p

=

po(l

- p/pe)-/ (where fo is the slip length at low shear rates,

j denotes the shear rate, and )c is a constant), suggesting that the slip length is governed by some kind of critical phenomenon. Other notable work [81, 82] has instead established that slip exhibits many of the hallmarks of a thermally activated process, at least for simple fluids in contact with atomically smooth boundaries. It has been proposed that fluid structuring in the vicinity of the fluid-solid interface plays an important role in determining the slip length in the fluid; in particular, it has been shown that, under some conditions, the slip length is inversely proportional to the density of the first fluid layer at the interface [78].

(29)

Fluid sliD at solid interfaces has also been investigated experimentally [4, 3, 5].IY L

These experiments have revealed several qualitative trends about fluid slip, but their results are not collectively well described by any existing model. Thus, despite some progress, complete and predictive models of fluid slip based on molecular considerations have yet to be developed.

1.3

Thesis Overview

In this thesis, we present our progress in theoretical and computational modeling of nanoscale fluid structure, transport, and structure-transport relationships. In Chapter 2, we present a series of results that illuminate the origins of fluid layering at a rigid solid interface. Using molecular mechanics and statistical physics, we identify and model the length and density scales that characterize fluid layering, in systems of semi-infinite extent. We also introduce the Wall number (Wa), a dimensionless group that governs the extent of fluid layering. We also present a universal scaling relation that relates layering in dense fluids to temperature. We finish by discussing the robustness of these results for describing systems of finite size and systems featuring different models for the solid.

In Chapter 3, we explore the connections between fluid layering and anoma-lous diffusion under nanoconfinement. We demonstrate that within fluid layers, diffusion is generally significantly slower than diffusion within the bulk fluid. We show that these slower dynamics are due to two primary causes: higher density and a restriction in the number of accessible dimensions. Using ther-modynamics and statistical physics, we are able to predict the extent to which diffusivity is hindered within the near-wall region. We also show that the Wall number can be used to predict how much of the fluid in a nanoscale system will

(30)

exhibit anomalous diffusion, which allows us to predict the overall diffusivity in a nanoscale system as a function of the confining length scale.

In Chapter 4, we tackle the notoriously difficult problem of dense fluid slip at solid interfaces. We propose that the molecular origin of slip is thermally acti-vated hopping of interfacial fluid molecules. Using molecular-kinetic principles, we develop a model that relates the slip velocity to the fluid-solid interaction strength, temperature, degree of fluid layering at the interface, and shear rate. We show that this model recovers the well known Navier slip condition in the limit of low shear rate. We also extensively discuss the close agreement between this model and previous slip theories, simulations, and experiments.

We conclude in Chapter 5 by summarizing the key contributions of this thesis and offering perspectives on future directions for this work. In Appendix A, we briefly outline the fundamentals of MD simulation and describe in detail the

(31)

Chapter 2

Fluid Structure at a Solid Interface

"The longer you are in a place, the more you get under its layers."

- Frances Mayes

Near a fluid-solid interface, the fluid spatial density profile is highly non-uniform at the molecular scale. This non-non-uniformity can have profound effects on the dynamical behavior of the fluid, and has been shown to play an especially important role when modeling a wide variety of nanoscale heat and momentum transfer phenomena. The critical role of the interfacial fluid structure strongly motivates the need for models that can describe and predict its characteristic length scales and density variations. Although the interfacial fluid structure typically exhibits several pronounced fluid layers (depending on factors like the fluid-solid interaction strength or the temperature), this layering effect decays as one moves away from the interface. As a result, in this chapter, we will focus on the first fluid layer, which is the most distinctly defined and most influential for anomalous interfacial phenomena; we will refer to this layer as the "first layer." Although interfacial fluid structure can be accurately described using liquid-state density-functional theory [29], our focus here is on developing

(32)

models of significantly lower complexity that highlight the basic molecular-mechanical factors governing the structure of the first layer. In particular, we will use molecular mechanics arguments and MD simulations to develop a better understanding of the structure of the first layer in the "layering regime," which is delineated by a non-dimensional number that compares the effects of wall-fluid interaction to thermal energy.

This chapter is organized as follows: In Section 2.1 we study the structure of the first fluid layer in semi-infinite systems using MD simulations and molecular mechanics arguments. We introduce a non-dimensional number which identi-fies the conditions under which significant layering is observed and develop molecular models for the width and number density associated with the first fluid layer. In particular, using asymptotic analysis of the Nernst-Planck equa-tion, we show that features of the fluid density profile close to the wall, such as the areal density of the first layer, .FL (defined as the number of atoms in this layer per unit of fluid-solid interfacial area), can be expressed as polynomial functions of the fluid average density, pave. This is found to be in agreement with MD simulations which also show that the width of the first layer, hFL, is a linear function of the average density and only a weak function of the temperature

T. We also demonstrate how these results can be combined to show that, for

system average densities corresponding to a dense fluid (Pave > 0.7), the density enhancement C(T) =aL1FL obeys a universal scaling independent of fluid den-sity. In Section 2.1.4, we will describe a complementary approach to quantifying the first layer density, which is based on geometric and packing considerations. In Section 2.2 we show that our results for infinite systems also hold for finite systems as well as systems under nanoconfinement, provided the average fluid density is appropriately defined. In Section 2.3, we show how the results of this work can be generalized to a broader class of fluids and solids. This chapter is

(33)

based in large part on work published in [83].

2.1

Fluid Layering in Semi-Infinite Systems

We begin by considering one of the simplest systems that can exhibit fluid layering, namely, a simple fluid at an atomically smooth solid boundary. In particular, we study a fluid whose atoms interact via a Lennard-Jones (LJ) [84] potential

u*(r) = 4 Uff (r [E r / r*6r \ r6 (2.1) The fluid rests in a slit-like geometry, bounded in the z direction by two solid boundaries, or walls. The distance between the two boundaries is denoted by

L*. The interaction between the fluid and the boundaries is also of the LJ type

with parameters E* and a* and is denoted by u*. Throughout this chapter,

stars denote dimensional quantities; unless otherwise stated, non-dimensional quantities are scaled by the LJ length scale a*, energy scale C, and temperature scale E*/kB, where kB is Boltzmann's constant.

For the majority of the MD results presented in this work, the solid bound-aries are composed of one graphene sheet held rigid with periodic boundary conditions in all directions; the effect of multiple rigid graphene layers as well as non-rigid walls is discussed in Section 2.3. Full details on the MD simulations can also be found in Appendix A.

We begin the investigation by considering the semi-infinite fluid case, where L is sufficiently large that a well defined bulk-fluid region separates the fluid-solid interfaces, so that the two interfacial regions have negligible effect on each other. In practice, L > 30 suffices; in our simulations, L = 32. We discuss finite systems

(L < 30) in Section 2.2. Our simulations are performed within the temperature 33

(34)

range 0.6 < T < 20 and within the average density range 0.4 Pave 1.2. Here Nttr3

Pave Vacc (2.2)

where Vacc is the volume accessible to the fluid; this quantity is more precisely de-fined in the next section. For semi-infinite systems, we also use the symbol PbuIk

(the "bulk density") interchangeably with Pave, since density inhomogeneities in the vicinity of the solid boundary only have a negligible effect on the channel-averaged density i.e. the channel-channel-averaged density is essentially equal to the density of fluid far away from the solid. The models that we present are valid within the layering regime i.e. conditions under which distinct fluid layers form, which we will now discuss.

2.1.1

The Wall Number and its Thermodynamic Interpretation

The concepts of layering that we develop in the following sections presume the presence of at least one distinct layer i.e. a pronounced first peak in the fluid spatial density profile. Generally speaking, layering occurs when the wall-fluid interaction is relatively strong. In particular, layering occurs when the energy scale of wall-fluid interaction in the interfacial region is large compared to the energy scale of thermal motion (which generally tends to discourage the formation of ordered structures). Given the above, and approximating the wall as an infinite plane with a density of n* atoms per unit area, we define the Wall number

Wa .

n*o(3)

kBT*

The Wall number is a measure of the relative importance of wall-fluid interaction energy (oc un*a*2L*) to thermal energy (oc kB3T*). From its definition, we expect that

(35)

Wa < 1 will indicate relatively small ig in the spatial density profile

near the wall, whereas Wa <#: 1 will correspond to the presence of well defined layering. As stated above, this chapter focuses on the strongly layering regime denoted by Wa <,: 1. Given the non-dimensional set of units used throughout

this chapter, we note that, in these units, Wa =, where n = n*-*2

We can readily develop a straightforward thermodynamic interpretation and intuition for the Wall number. At any temperature, the condition of thermody-namic equilibrium for the wall-fluid system is the minimization of the Helmholtz free energy F = U - TS, where U is the internal energy and S is the entropy. At

low T (high Wa), the system's only way to minimize F is to minimize U, which tends to encourage layered (periodic) structures that feature fluid atoms resting in the potential-energy well generated by the solid. But as T increases (Wa de-creases), the minimization of F increasingly relies upon the maximization of S. For a fixed particle number and channel width, the entropy-maximizing fluid distribution is the uniform distribution (no layers), and so the low-Wa regime corresponds to negligible fluid layering.

2.1.2

The Degree of Layering and the Layering Regime

To quantify the notion of a well defined first layer, we now define the degree of

layering as

max* p(z*)

-L = mic: '2-PZ)(2.4)

where p(z*) is the fluid spatial density profile in the direction orthogonal to the fluid-solid interface. This definition is schematically illustrated in Figure 2-1. From MD simulations, we find that _ > 5 is sufficiently pronounced layering for the purposes of our modeling below. In Figure 2-2, we can clearly see that the degree of layering increases dramatically with increasing Wall number. In

(36)

particular, the layering regime X > 5 corresponds to Wa 4z 1.

C

a)

max

p z

/

max

p z

Distance from solid interface

Figure 2-1: Schematic illustrating the definition of the degree of layering X, which is calculated as the ratio of the magnitude of the first peak on the fluid density profile (shown in green) to the magnitude of the first minimum between the first two peaks (shown in orange).

2.1.3 Anatomy of the First Layer: Length Scales of Layering and

the Density Enhancement

Figure 2-3 shows the distinct layering of fluid near the fluid-solid interface for systems with Wa 4 1. Characteristic fluid density profiles in the vicinity of the

(37)

8

6

O 4

c,)

0

2

0

*

*

*

I

-*

*

*

*

I

* MD Simulation

0

2

4

Wa

6

8

Figure 2-2: Relation between degree of layering and Wall number, obtained from 33 MD simulations of

J

fluid confined within a graphene nano-slit with

0.4 s Pave ! 1.2 and 0.6<s T s 20.

(38)

wall are shown in Figure 2-4. From this figure we can see that, in the strongly layering regime, density variations extend up to t, ~ 5 from the solid and more than five distinct layers are discernible before the density relaxes to the bulk

value pbulk Of particular interest here are the stand-off distance between the wall and the fluid, denoted by Zmin, and the thickness (width) of the first layer, denoted hF; both are shown in Figure 2-4. An additional quantity of interest is the overall particle content of the first layer. These quantities are defined more precisely and discussed in more detail in the sections that follow.

Ito0

Figure 2-3: Fluid layering near the graphene surface, obtained from MD

sim-ulation at T = 6.4 and pave = 1.2 (Wa = 0.6). The minimum separation zmi is

schematically illustrated in orange.

Stand-off Distance of the First Layer

The stand-off distance between the fluid and the wall is primarily a result of the repulsive interaction between the two wall and the fluid. As we will explain in

(39)

6 5 o C 0) a N E

z

1 :

..-.. Low Density, Low Temperature S...Low Density, High Temperature

- - High Density, Low Temperature

* I- - High Density, High Temperature

FL

13

0 0.5 1 1.5 2 2.5 3

Distance from Wall

3.5 4 4.5 5

Figure 2-4: Four smoothed fluid spatial density profiles, corresponding to low density (Pave = 0.5) and high density (pave = 1.0) as well as low temperature

(T = 3.2) and high temperature (T = 6.4). In all cases, density variations are

low in amplitude at a distance of 5a from the wall. The first-layer width hFL

is schematically illustrated in green. Note that hFL is a function of density, and also a weak function of temperature in the low-density case. The minimum separation Zmin given in Equation (2.8) is schematically illustrated in orange. Note that this quantity does not discernibly depend on density or temperature.

greater detail in Section 2.1.4, a closed-form expression for Zmin can be obtained

by considering the competition between the wall-fluid interaction and thermal

energy, using a mean-field theory approach described in [51]. For the system studied here, this theory predicts Zmin = 0.86 (Figure 2-5), independent of density or temperature. Our MD results, shown in Figure 2-5, verify that the mean-field theory prediction is valid for a wide range of temperatures (0.6 T 20) and fluid densities (0.4 < Pave 1.2).

As will be seen below, this result has a number of implications. Here we discuss its application to the accessible volume defined above. In a graphene

(40)

nano-slit geometry, this quantity is given by Vacc = (L - 2zmin)A, where A is

the interfacial contact area between the fluid and the solid. Accounting for the excluded volume due to Zmin becomes particularly important in finite systems, where this volume is appreciable compared to the total system volume, as further

discussed in Sec. 2.2.

1

0.8

E

N

0.6

0.4

0.2

0

0.4

0.6

0.8

1

1.2

Pave

Figure 2-5: MD results (0.6 T 20) for the stand-off distance zmin are shown as a function of density, along with the constant value predicted by mean-field theory in Equation (2.8). Note that MD measurements of Zmin are nearly independent of the temperature and fluid density.

*

MD

Simulation (T = 0.6)

*

MD

Simulation

(T

=

5)

*

MD

Simulation

(T

=

10)

*MID Simulation (T = 15)

-

*

MID Simulation (T = 20)

Theory

(41)

First-Layer Width

We define the first-layer width, denoted by hFL, as the distance between Zmin and the first non-zero minimum of the density profile. This distance is shown schematically in Figure 2-4. To prevent small density fluctuations over space from materially affecting the calculation of the first non-zero minimum, we per-form moving-average smoothing on the density profiles obtained from MD, with a window size of 0.5. By studying window sizes between 0.3 and 1.1, we have verified that the location of the first non-zero minimum is not meaningfully affected by the choice of the window size (the location varies by less than 5% depending on window size). We note here that, unlike Zmin, hFL is not neces-sarily independent of the temperature and fluid density; these dependences are discussed in greater detail in Section 2.1.4.

The Density Enhancement

Having established the length scales relevant to the first layer, we now define the first-layer density enhancement, a measure of the particle content of the first layer, as

C =PFL (2.5)

Pave

where PFL is the (volumetric) density of fluid molecules contained within Zmin 5

Z Zmin + hFL. Although it is reasonable to expect that, in general, C = C(p, T)

-for example, the magnitude of the density peaks appear to be quite sensitive to the system density and temperature (see Figure 2-4) - we also expect that

lim C(p, T) = 1 (2.6)

T -+ai

based on the thermodynamic arguments developed in Section 2.1.1.I

(42)

Figure 2-6 illustrates the variation of the density enhancement with temper-ature in MD data (along with predictions from models that we will develop in subsequent sections). Surprisingly, Figure 2-6 shows that for Pave 0.7, C is

essentially independent of density not only for T -+ wc but also for T > 6. More-over, its temperature dependence is very weak. This remarkable relationship implies that the fluid density in the first layer, based on the width hFL, is related to the average fluid density via a density-independent enhancement factor that is only weakly dependent on temperature.

2.1.4

Molecular Mechanics of the First Layer

In this section we use MD simulations and molecular mechanics modeling to explain the behaviors observed in Section 2.1.3). In addition to the stand-off distance zmin and enhancement factor C, we also define and investigate the behavior of the fluid areal density in the first layer, denoted by EFL- In particular, we show that a theoretical framework based on the Nernst-Planck equation can be used to generate several significant insights into the variations of these structural features with respect to temperature and fluid density.

Stand-off Distance between the Wall and the First Layer

The first fluid layer forms at a characteristic distance from the wall that can be calculated using the mean-field approach proposed in [51]. Assuming that the solid surfaces are sufficiently large such that edge effects are negligible, the mean-field interaction potential between the solid and a fluid atom at distance z

(43)

10 T 20 * pave = 0 4 * pave = 0.5 Save

=

0.6

*

ave 0 7

*

pave 0 Pave = 0.8 P Pave

=

0.9

*

Pave 1 0

*

pave 1

-*

Pave 1 2 - Using F,

.

Using

s-2

4

6

8

10

12

14

16

T

Figure 2-6: Density enhancement as a function of temperature for 0.7 < Pave

1.2 (inset shows the density enhancement for three additional low densities, 0.4 < Pave 0.6). Note that the density enhancement decreases with increasing temperature for all densities. Moreover, for Pave > 0.7, the enhancement can be accurately modeled using EFL from Equation (2.28) (represented by dashed curve) as well as using s-2from Equation (2.36) (represented by diamonds).

43 4

3.5

3kp

u

2.5 f

* ** 4 3.5 3 : 2.5 24 1.5 1 0

2

1.5

1

2

(44)

each atom within the solid, which yields:

Ulane(z*) = 2n5*n 2 -- 5 (2.7)

As before, n denotes the (non-dimensional) areal density of solid atoms in the graphene sheet. This mean-field potential rises sharply as z -+ 0, making regions

close to the solid described by z* < z; inaccessible to the fluid. The value of Zmin can be accurately estimated by setting U*n(z*) = 0. The rationale for this

choice is that the rise of U,(z*) is so sharp that solving for Ua (z*) = K, where K

is some constant on the order of the thermal energy (K ~ kBT*), leads to a more cumbersome but for all practical purposes equivalent result. Proceeding, we find

Zmin = (216 (2.8)

This result is within 8% of the stand-off distance observed in Sec. 2.1.3 for 0.4 : pave 1.2 and 0.6 T 20. We also note that (2.8) is identical to the result obtained in [51] for the stand-off distance in a cylindrical geometry (e.g. carbon nanotubes). This is because the effects of curvature do not appear in the leading-order calculation presented in [51].

Areal Density of the First Layer

Although the first layer is not strictly a two-dimensional structure (since it has a characteristic width of hFL), this layer tends to be narrow. This observation motivates us to characterize its packing using an areal density defined by the relation

EFL PFLhFL CpavehFL (2.9)

(45)

The area] density is, of course, related to the spatially varying particle number

density p(z) by the integral relation

min+hFL

EFL JZmin p(z)dz (2.10)

Zmin

In principle, EFL can be calculated from solution of the Nernst-Planck equa-tion [85]

d2p* d ( p* dU*

dZ2 dz kBT* dz(2.11)

Here, the total potential energy per particle is written as

U*(z) = U;,(z*) + U*(z*, p*(z*)) (2.12) where the potential energy due to the solid-fluid interaction is denoted by

U(z*) and the potential energy due to the fluid-fluid interaction is denoted

by U*(z*, p*(z*)). These two quantities can be calculated from

U,(z*) = nj u* [r*(z*, x*,y*/,z*')]dQ*,(x*',y*',z*/) (2.13)

and

U*(z*) = u* [r*(z*,x*',y*',z*')]p(z*')dQ*(x*',y*',z*') (2.14)

Pave E uf f[r(z, x', y', z')] p(z')dQf (x', y', z')

where D), and Of denote the wall and fluid domains, respectively, and p =

P/Pave. We note that Equation (2.13) simply evaluates to the result of Equation

(2.7) for the case of the graphene walls studied in this work. From the above

(46)

two expressions we see that U* - nE*, while U* ~ paveE*.

Equation (2.11) can be integrated once to yield,

dp * p* dU*

dz* kBT* dz*

This equation needs to be solved subject to the constraint of fixed particle number in the system,

IL'

0

p*(z*)dz* = p* *.

Taking into consideration the dominant role of the solid boundary in the vicinity of the solid-fluid interface, we write

U,(z*) _ n kBT* -U,(z) U* (z*, p*(z*)) kB T* Pave E* -*T Uf (z*, P(z*)) T eUf (z*pz)

Here, we have introduced a small parameter e defined as

e = paveE (nE*).

Equation (2.15) can thus be written as

- = -Wap

duw

dZ ( dz +

dU(P))

Without further analysis, we can immediately glean two insights. In the low-Wa regime (Wa< 1), Equation (2.20) implies that variations of the density are negli-gible i.e. the density is nearly constant, as expected from our earlier discussion in Section 2.1.1. On the other hand, if Wa4: 1, then the form of Equation (2.20)

(2.15) (2.16) (2.17) (2.18) (2.19) (2.20)

(47)

generally suggests that variations of the density are oscillatory, with the

ampli-tude (closely related to the degree of layering) modulated by the magniampli-tude of Wa.

Delving further, based on the structure of Equation (2.20) and the observation that typically e < 1 (in our simulations e ~ 0.2), we propose a solution of the form

00

(2.21)

recalling that e oc Pave.

Using Equation (2.21), we find to zeroth order

dj-o dU,

-- =

-waj

dz dz

while at the next order we find

=-Wap* - WaPO. dUf(o)

dz 1 dz dz

Equation (2.22) can be solved directly using the constraint

I

L podz = L.

(2.22)

(2.23)

(2.24)

Subsequently, given po, Equation (2.23) can be solved subject to the constraint

I

L

p

1dz = 0. (2.25)

In principle, we can extend this approach to solve for the fluid density profile associated with each order of e. In practice, however, a numerical treatment is

(48)

required due to the notorious complexity associated with evaluating Uf(p). At this point, we note that "exact" solutions (to all orders of e) of the above problem are available via MD simulation. Here, we will use the above Nernst-Planck solution framework to make some general deductions in support of our MD simulation results. In particular, from the above discussion we can see that

p(z) = Pave Po(Z) + ep1(z) + O(C2)(. (2.26)

This confirms that in the regime {Wa - 1, e < 1} features of the fluid local

density can be described in polynomial expansions of the average fluid density. In other words, we expect that

EFL(Pave,T) aj(T)pve (2.27)

j=1

To validate the above result, we conducted MD simulations over a range of densities and temperatures such that Wa #$ 1 and e < 1. As shown in Figures 2-6 and 2-7, we find that our MD results can be described quite accurately in the

range 0.4 < Pave < 1.2, 2 T 15 using the form

EFL(Pave, T) = al(T)Pave + a2(T)pve + a3(T)P3ve (2.28)

where

a1(T) = 0.994 + 4.797T 1 = 0.994 + 1.267 Wa = a1(Wa)

a2(T) = 0.071 - 8.738T-1 = 0.071 - 2.307 Wa = a2(Wa)

a3(T) = -0.258 + 4.431T-1 = -0.258 + 1.170 Wa = a3(Wa)

Figure

Figure  1-1:  Visualization  of equilibrium Lennard-Jones  fluid  structure within  a CNT  (R  =24  A)  obtained  from MD  simulation
Figure  1-2:  Representative  radial  density  function  for  fluid  confined  within  a CNT  at 300  K,  including  a sharp  density peak at  a distance  of approximately  3
Figure  2-1:  Schematic  illustrating  the  definition  of  the  degree  of  layering  X, which is  calculated  as  the  ratio  of the magnitude  of  the first  peak on the  fluid density profile  (shown in green) to the magnitude of the first minimum betw
Figure  2-2:  Relation  between  degree  of  layering  and  Wall  number,  obtained from 33  MD simulations  of  J  fluid confined  within  a graphene  nano-slit with 0.4  s Pave  !  1.2  and 0.6&lt;s  T  s 20.
+7

Références

Documents relatifs

Abstract : This study is devoted to the calculation, in transient mode, of the ultrasonic field emitted by a linear array and reflected from a fluid-fluid interface thanks to a

2shows the reflected pressure at a fixedheight h=2mm from the interface and at two different axis positions x=0mm and x=8mm.Differentpulses results, they corresponds to the direct

Thus the dependence of the dynamic contact angle on the three-phase line velocity appears to be frequency dependent above I Hz in our experiment. Moreover, the fact

convincing figures showing sections through the ditch (fig. 52) and questions relating to the nature and origin of the rubbish layer predating the use of this area as a military

On a remarqué que les données exposent des localités différentes et le cache de données unifié n’exploitent pas les localités exhibées par chaque type de

In fact, the Smirnov distribution of total amounts of time spent in the bulk af'ter n steps, equation (21), can provide us with such an expression for r (r, t if we make use of the

Whereas for submicellar solution, common non-ionic surfactants molecules usually undergo a diffu- sion limited adsorption, we showed that the adsorption of free surfactants from

(line + symbol) for five aerosol components at 532 nm; (b) extinction Ångström exponents at 355–532 nm obtained from lidar observations and modeled by MERRA-2 for pure dust