Bhushan B. Handbook of Micro/Nano Tribology, Second Edition

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Other studies have concentrated not on the formation of adhesive bonds, but on the process by which

these bonds are broken as two surfaces are separated. The failure of adhesive bonds is a complicated

phenomenon that involves both interfacial attraction and kinetic effects. Energy dissipation mechanisms

play a dominant role in the function of adhesives because the mechanical energy required to break a

bond, G, can be 10

times greater than the reversible work of adhesion, W. Thus, the excess work, G –

W, is dissipated during rupture of the adhesive.

Baljon and Robbins (1996, 1997) used MD simulations to follow the movement of individual atoms

and energy dissipation during the rupture of a thin adhesive ﬁlm. The model system contained two rigid

solid walls joined by a thin adhesive ﬁlm. In separate studies, ﬁlms composed of linear-chain molecules

of between 2 and 32 monomers (Figure 11.15) were investigated. The monomers interacted via a trun-

cated LJ potential and adjacent monomers along each chain were also coupled through an attractive

potential that prevented chain crossing and breaking (Kremer and Grest, 1990). The rigid solid walls

consisted of two (111) planes of an fcc lattice. The wall atoms were coupled to lattice sites by stiff springs

and their nearest-neighbor spacing was ﬁxed at 80% of the equilibrium monomer spacing. The temper-

ature was kept constant by coupling the wall atoms to a heat bath. For the results discussed here, the

chains were composed of 16 monomers, the adhesive contained 2048 monomers, and each wall consisted

of 800 atoms. Rupture was simulated by separating the walls with a uniform velocity. The particle motions,

forces, potential and kinetic energies, and heat ﬂow toward the bath were all monitored throughout the

course of the simulation.*

The excess of work G – 2γ (γ is the surface tension) as a function of rupture velocity and the shear

response (friction) vs. shear velocity were examined at three different temperatures (Figure 11.16). The

excess work and the shear response exhibited different behavior in each temperature regime. In the low-

FIGURE 11.15 Snapshots of a glassy ﬁlm during rupture

at ν = 0.003σ/τ, where σ and τ are characteristic length

and timescales of interaction. The 800 atoms in each wall

are black, and monomers of the 128 chains are colored.

Different colors were used to make the chains forming the

bridges visible. The vertical direction corresponds to the

z-direction and periodic boundary conditions were applied

in the xy plane. Only a thin cross section of the ﬁlm is

shown in the second panel from the top. (From Baljon

A. R. C. and Robbins, M. O. (1997), Mater. Res. Soc. Bull.

22, 22–26. With permission.)*

* Color reproduction follows page 16.

temperature glassy state, the excess work and friction approached constant values at low velocities, v.

These limits correspond to the amount of adhesion hysteresis and static friction, respectively. Just above

the glass-transition temperature both excess work and friction increased as v

, where x = ⅓. At lower

velocities and higher temperatures, there was a newtonian regime in which both quantities increased

linearly with velocity.

Above the glass-transition temperature, rupture produced smooth viscous ﬂows in the ﬁlm that

resulted in a correspondence between the excess work and the shear stress. In addition, both ﬂow velocities

and dissipation approached zero at low velocities. In contrast, there was no correspondence between the

excess work and the shear stress in glassy ﬁlms. The shear was conﬁned to the region between the ﬁlm

and the wall, and rupture occurred in the ﬁlm through a sequence of rapid structural rearrangements.

As the glassy ﬁlm was separated, the ﬁlm deformed elastically, and the force needed to separate the

walls and the distances between layers increased linearly with time (Figure 11.17). Eventually, a threshold

layer separation was reached and the ﬁlm became unstable against density ﬂuctuations. Small cavities

formed and grew, allowing the remainder of the ﬁlm to relax to its original density. The interlayer

separation where cavities formed was found to be independent of chain length and, thus, depended only

FIGURE 11.16 Excess work as a function of rupture velocity

(upper panel) and mean frictional force on the tip wall as a

function of shear velocity in the (100) direction at zero pressure

(lower panel). Lines with slope

⅓ (dashed) and 1 (solid) indi-

cate the power law scaling observed at T = 0.6 ε and 1.1 ε,

respectively. (From Baljon, A. R. C. and Robbins, M. O. (1996),

Science 271, 482–484. With permission.)

FIGURE 11.17 Time dependence of energy changes ∆E

(upper panel) and the force on the walls (lower panel) during

rupture of the glassy ﬁlm shown in Figure 11.15. Lines in the

upper panel indicate the external work (long dashes), the

potential energy increase (short dashes), and the total heat

ﬂow (solid). Because the mean kinetic energy remains con-

stant at ﬁxed temperature, the work equals the sum of the

potential energy change and heat ﬂow. (From Baljon, A. R. C.

and Robbins, M. O. (1996), Science 271, 482–484. With permis-

sion.)

on the force between individual monomers. Each time the internal stress in the ﬁlm exceeded the local

yield stress, further increases in the wall separation resulted in sudden structural rearrangements of the

ﬁlm (Figure 11.15, second panel from top). In the latter stages of rupture, the cavities coalesced

(Figure 11.15, second panel from bottom). The lengths of the bridges connecting the walls grew to nearly

that of a fully stretched chain before one end of the chain pulled free and collapsed onto the opposite

surface (Figure 11.15, bottom panel). The ﬁnal surfaces were very rough. Energy stored in the excess

surface area was part of the unrecoverable work, which was gradually converted to heat as the surface

annealed.

By examining hysteresis loops, the authors determined that the excess work was dissipated evenly

among cavitation, plastic yield, and bridge rupture. All of these processes dissipated more energy with

increased ﬁlm thickness and chain length. The adhesive energy G for glassy ﬁlms was approximately twice

the reversible work.

In a similar type of study, Streitz and Mintmire (1996) studied the elastic and yield response of bulk

and thin-ﬁlm α-alumina. The technological importance of this material at metal–metal oxide interfaces

was the driving force behind its selection. A thin ﬁlm of α-alumina was constructed from a slab ten layers

thick of α-alumina (about 25 Å thick) with two free (0001) surfaces. Periodic boundary conditions were

applied in the plane of the free surfaces, but not normal to the surfaces. The forces between atoms were

modeled using a variable-charge electrostatic model plus an EAM potential (ES+ method) (Streitz and

Mintmire, 1994). The ﬁlm was equilibrated; then a sequence of strains was applied by moving the

outermost layers of the ﬁlm a speciﬁed amount normal to the free surfaces. The remainder of the atoms

were allowed to relax, and at each subsequent increment of total strain the internal atoms were allowed

to reach equilibrium.

Streitz and Mintmire (1996) calculated a stress–strain curve for the application of compressive and

tensile strains to both the bulk and thin-ﬁlm systems. The slab responded elastically during initial loading

and unloading, and the appropriate elastic constant could be calculated from these data. In addition, the

variation of the elastic constant c

was also examined as a function of strain. In general, c

(the modulus)

increased and the material became stiffer for compressive strains, while for tensile strains the modulus

decreased and the material became softer. For the bulk system, c

varied approximately linearly with

strain, while marked deviations from linearity were apparent for the thin ﬁlm. The value of c

at zero

strain was calculated to be 498 GPa for the thin ﬁlm, in close agreement with the bulk calculation that

yielded 509 GPa. Last, a theoretical yield stress of 44.5 GPa was calculated. This value is not far outside

the theoretically determined range of 35 to 40 GPa. Thus, the authors concluded that the ES+ method

is capable of describing the elastic response of α-alumina in situations that are far from thermodynamic

equilibrium. In addition, these simulations showed that the elastic response of the α-alumina as a thin

ﬁlm might differ substantially from the elastic behavior of a bulk crystal. This conclusion is quite relevant

in view of the prominent use of α-alumina as a coating.

11.4 Lubrication at the Nanometer Scale:

Behavior of Thin Films

Experiments have shown that the properties of ﬂuids conﬁned between solid surfaces are drastically

altered as the separation between the solid surfaces decreased (Horn and Israelachvili, 1981; Chan and

Horn, 1985; Gee et al., 1990). For instance, at separations of a few molecular diameters, liquid viscosities

increase by several orders of magnitude (Israelachvili et al., 1988; Van Alsten and Granick, 1988). Con-

tinuum hydrodynamic and elastohydrodynamic theories, which have been successful in describing lubri-

cation by micron-thick ﬁlms, begin to break down when the thickness of the liquid approaches the

thickness of a few molecular diameters. Because an increasing number of applications involve lubricants

in such conﬁned geometries, the need to understand this sort of system through modeling has become

increasingly important.

Molecular dynamics simulations have begun to ﬁll the void created by the breakdown of continuum

theories. These simulations have revealed a number of new phenomena, several of which have explained

experimental observations pertaining to the behavior of conﬁned ﬁlms. The equilibrium properties of

ﬁlms of various types, such as spherical molecules, straight-chain alkanes, and branched alkanes conﬁned

between solid parallel walls have been examined. Spherical molecules, for example, have been shown to

order both normal and parallel to the solid walls. Film properties, such as viscosity, have also been

examined. Finally, while macroscopic experiments are consistent with one of the fundamental assump-

tions of newtonian ﬂow, namely, the “no-slip” boundary condition (BC), recent microscopic experiments

are not. (The “no-slip” BC requires that the tangential component of the ﬂuid velocity be equal to that

of the solid surface.) Therefore, ﬂow BCs have been examined using MD simulations (Thompson and

Robbins, 1990a,b; Thompson et al., 1992; Robbins et al., 1993). Some of the more recent work in these

areas is discussed in the following sections.

11.4.1 Equilibrium Properties of Conﬁned Thin Films

A number of groups have studied the equilibrium properties of spherical molecules (interacting through

LJ potentials) conﬁned between solid walls using both Monte Carlo methods (Schoen et al., 1987) and

MD simulations (Bitsanis et al., 1987; Thompson and Robbins, 1990b; Sokol et al., 1992; Diestler et al.,

1993). These studies have demonstrated that, irrespective of the atomic-scale roughness of the pore walls,

when a ﬂuid of spherical particles is placed inside a pore, the ﬂuid layers are layered normal to the pore

walls (Bitsanis et al., 1990).

The typical signature used to identify the ordering of the liquid is the liquid density plotted as a

function of distance from the pore walls (termed a density proﬁle). For example, Thompson and Robbins

(1990b) used MD simulations to examine the structure of LJ liquids conﬁned between two solid walls

that consisted of (001) planes of an fcc lattice. The simulation geometry (a slit pore consisting of two

parallel solid surfaces) was chosen to closely resemble an SFA (Figure 11.18a). Oscillations in the calcu-

lated density proﬁles corresponded to well-deﬁned, liquid layers (Figures 11.19a to c). (In Figure 11.19

the center of the pore corresponds to a z/σ value of zero and the distance z has been normalized by the

characteristic LJ length parameter σ.) In the middle of the pore no oscillations in the density proﬁle were

present; thus, the liquid possessed the unstructured density appropriate for a bulk liquid in this region

of the pore.

FIGURE 11.18 Schematic representation of the sim-

ulation geometry used to model the conﬁnement of

liquids between parallel solid walls. Projections of liquid

particle positions on the xz plane are represented as

open circles and the wall molecules as ﬁlled circles in

(a). (From Thompson P. A. and Robbins, M. O. (1990),

Phys. Rev. A 41, 6830–6837. With permission.) There

are 672 ﬂuid atoms and 192 wall atoms. The walls are

moved at a constant velocity U and in opposite direc-

tions along the x-axis. A sketch of a slightly different

simulation geometry is shown in (b). The walls are held

together by a constant load P

⊥

. The upper wall is

attached by a spring to a stage that is moved at constant

velocity v. (From Thompson, P. A. and Robbins, M. O.

(1990), Science 250, 792–794. With permission.) Peri-

odic boundary conditions are imposed in the xy plane.

Normal ordering of the ﬂuid and even a phase transition to a solid structure where layers of the liquid

become “locked” to the solid walls can be induced by increasing the strength of the interaction between

the walls and the ﬂuid (the ratio ε

/ε in Figure 11.19) (Schoen et al., 1989; Thompson and Robbins,

1990b; Sokol et al., 1992). This phenomenon manifests itself as larger oscillations in the density proﬁles

(Figure 11.19b and c). This effect was also observed when n-alkanes were trapped between structured

walls (Ribarsky and Landman, 1992).

Schoen et al. (1987) were the ﬁrst to observe that structure in the walls of the pore induces transverse

order (parallel to the walls) in a conﬁned atomic ﬂuid. Using grand canonical MD studies of an atomic

ﬂuid conﬁned between fcc (100) planes of like atoms, they demonstrated that for pore thicknesses of

approximately 1 to 6 atomic diameters, the ﬂuid alternatively freezes and thaws as a function of pore

thickness. The solid formed epitaxially in distorted fcc (100) layers. This epitaxial effect decreased with

increasing pore thickness but persisted indeﬁnitely in the layer nearest to the pore wall. In a related work,

a detailed analysis of the structure of the ﬂuid within a layer, or epitaxial ordering, as a function of wall

densities and wall–ﬂuid interaction strengths was undertaken (Thompson and Robbins, 1990b). For

small ratios of wall-to-liquid well depth (ε

/ε = 0.4), ﬂuid atoms were more likely to sit over gaps in

the adjacent solid layer; however, self-diffusion within this layer was approximately the same as in the

bulk liquid. In other words, although the solid induced order in the adjacent liquid layer, it was not

sufﬁcient to crystallize the liquid layer. Increasing the strength of the wall–ﬂuid interactions by a factor

of 4.5 resulted in epitaxial locking of the ﬁrst liquid layer to the solid. This epitaxial ordering was

conﬁrmed from an analysis of the two-dimensional structure factors, the spatial probability distribution,

mean-square displacement of the atoms within the layer, and the diffusion within the layer. While

diffusion in the ﬁrst layer was too small to measure, diffusion in the second layer was approximately half

of its value in the bulk ﬂuid. The second layer of liquid crystallized and became locked to the ﬁrst “liquid”

layer when the strength of the wall–liquid interaction was increased by approximately an order of

magnitude over its original value. A third layer never crystallized.

The conﬁnement of linear-chain molecules has also been examined by a number of groups (Ribarsky

and Landman, 1992; Thompson et al., 1992; Wang et al., 1993a,b). For example, using a simulation

geometry similar to that shown in Figure 11.18b, Thompson et al. (1992) examined the conﬁnement of

linear-chain molecules between two (111) fcc planes. The linear-chain molecules were modeled via the

bead-spring model, which has been shown to yield realistic dynamics for polymer melts (Kremer and

FIGURE 11.19 Liquid density ρ(z) and the xz component of the micro-

scopic pressure-stress tensor P

as a function of distance between the walls

for a number of wall ﬂuid interaction strengths (ε

/ε). (The wall velocity

U = 0.) The solid line in (d) represents P

averaged within the ﬂuid layers.

All quantities have been normalized using the appropriate variables so that

they are dimensionless. (From Thompson, P. A. and Robbins, M. O. Phys.

Rev. A 41, 6830–6837. With permission.)

Grest, 1990). Adjacent monomers were coupled via an attractive potential and non-nearest-neighbor

monomers interacted via a repulsive, truncated LJ potential.

Conﬁnement of the polymer between solid walls was shown to have a number of effects on the

equilibrium properties of the static polymer ﬁlms. The ﬁlm thickness decreased as the normal pressure

on the upper wall increased. At the same time, the degree of layering and in-plane ordering increased,

and the diffusion constant parallel to the walls decreased. In contrast to ﬁlms of spherical molecules,

where there was a sudden drop in the diffusion constant associated with a phase transition to an fcc

structure, ﬁlms of chain molecules remained highly disordered and the diffusion constant dropped

steadily as the pressure increased. This indicated the onset of a glassy phase at a pressure below the bulk

transition pressure. This wall-induced glass phase has provided a natural explanation for the dramatic

increases in measured relaxation times and viscosities of thin ﬁlms (Gee et al., 1990; Van Alsten and

Granick, 1988).

The conﬁnement of n-octane between parallel, crystalline solid walls was examined by Wang et al.

(1993a,b) using MD. A more realistic liquid potential energy function (Jorgensen et al., 1984) was used

and rigid Langmuir–Blodgett (LB) monolayers were used to model the walls of the pore (Hautman and

Klein, 1990). The pore was ﬁnite in one direction (typically 2.5 nm long) and made inﬁnite in the other

direction by the application of periodic boundary conditions. In this geometry, liquid exited the pore

and collected as a droplet in the ﬁnite direction (Figure 11.20). Liquid vapor from these droplets interacted

with vapors from the other side of the pore via the periodic boundaries (Figure 11.20a to f). The conﬁned

ﬂuid was in equilibrium with the bulk-like droplet at 1 atm and pore widths ranged from 1.0 to 2.4 nm.

For the smallest pore size examined (1.0 nm) the ﬁlm formed a layered structure with the molecules

lying parallel to the pore walls (Figure 11.20a). At larger pore widths (Figure 11.20b to f), there was

always a layered structure on each wall surface and more poorly deﬁned layers in the center of the pore,

with the exception of the 1.25-nm pore (Figure 11.20c). In that case, the ﬁlm ordered so that the alkane

molecules were oriented perpendicular to the walls. The oscillatory nature of the liquid density proﬁles

(Figures 11.21a to h) conﬁrmed the layered structure of the n-octane ﬁlms; computed diffusion coefﬁ-

cients for the ﬁlms, which were approximately equal to bulk values, conﬁrmed the liquid nature of the

ﬁlms.

The layering of these ﬁlms had profound effects on other equilibrium properties. For example, Wang

et al. (1993a) showed that the solvation force of n-octane thin ﬁlms increased dramatically as the pore

size decreased. Surface force apparatus experiments have also shown that the nature of the ﬁlm has an

effect on the solvation force. It is well known that linear alkane molecules tend to layer close to a surface.

This layering gives rise to oscillations in the density proﬁle (Christenson et al., 1989). While early

experiments indicated that the surface force oscillations vanish for branched alkanes such as 2-methy-

loctadecane (Israelachvili et al., 1989), more recent experiments (Granick et al., 1995) have shown oscil-

lations in the force proﬁles of branched hydrocarbon molecules containing a single-pendant methyl group

that are similar to those of linear hydrocarbons. Wang et al. (1993a,b, 1994) carried out MD studies on

conﬁned n-octane and 2-methylheptane and reached a similar conclusion.

In contrast, experimental studies that examined the conﬁnement of highly branched hydrocarbons

such as squalane showed that the surface force oscillations disappear (Granick et al., 1995). In an effort

to shed light on this, Balasubramanian et al. (1996) used both Monte Carlo and MD to examine the

adsorption of linear and branched alkanes on a ﬂat Au(111) surface. In particular, they examined the

adsorption of ﬁlms of n-hexadecane, three hexadecane isomers (6-pentylundecane, 7,8-dimethyltetrade-

cane, and 2,2,4,4,6,8,8-heptamethylnonane), and squalane. The alkane molecules were modeled using

the united atom approach with an LJ potential used to model the interactions between united atoms.

The alkane–surface interactions were modeled using an external 12-3 potential with the parameters

appropriate for a ﬂat Au(111) substrate (Hautman and Klein, 1989). The heptamethylnonane and

squalane ﬁlms were investigated using constant-NVT MD simulations. Other ﬁlms were examined using

conﬁguration biased Monte Carlo (Siepmann and McDonald, 1993a,b; Siepmann and Frenkel, 1992).

The Monte Carlo calculations yielded density proﬁles for n-hexadecane and 6-pentylundecane that

were nearly identical with experiment and previous simulations. In contrast, the density proﬁles of the

more highly branched alkanes such as heptamethylnonane and 7,8-dimethyltetradecane exhibited an

additional peak. These peaks arose from methyl branches that could not be accommodated in the ﬁrst

liquid layer next to the Au surface. That is, the branched hydrocarbons adsorbed in layered structures

with interdigitation of the molecules. For thicker ﬁlms, the oscillations in the density proﬁles for hep-

tamethylnonane were out of phase with those for n-hexadecane, in agreement with the experimental

observations (Granick et al., 1993).

Granick et al. (1993) did not observe force oscillations for squalane ﬁlms conﬁned between mica or

organic monolayers of octadecyltriethoxysilane when the surfaces of the SFA were separated by more

than 18 Å. In contrast, the MD simulations (Balasubramanian et al., 1996) yielded a density proﬁle for

the squalane ﬁlm that was very similar to the density proﬁle for 7,8-dimethyltetradecane. The most likely

reason for this discrepancy between theory and experiment was the fact that the ﬁlm was adsorbed, rather

than conﬁned, in the MD simulations.

FIGURE 11.20 Atomic conﬁgurations adopted by n-octane when conﬁned between parallel, rigid LB substrates of

various pore widths. The pore widths and liquid temperatures are 1.0 nm and 297 K in (a), 1.6 nm and 297 K in

(b), 1.25 nm and 297 K in (c), 1.25 nm and 250 K in (d), 1.8 nm and 297 K in (e), and 1.8 nm and 250 K in (f).

The light spheres represent ﬂuid (united) atoms and the dark spheres represent substrate atoms. (From Wang, Y.

et al. (1993), J. Phys. Chem. 97, 9013–9021. With permission.)

11.4.2 Behavior of Thin Films under Shear

The behavior of conﬁned spherical and chain molecular ﬁlms under shear has been examined using both

Monte Carlo methods (Schoen et al., 1987; Curry et al., 1994; Curry and Cushman, 1995a,b) and MD

simulations (see, for example, Schoen et al., 1989; Bitsanis et al., 1990; Bitsanis and Pan, 1993; Sokol

et al., 1992; Diestler et al., 1993; Thompson and Robbins, 1990b; Thompson et al., 1992; Curry et al.,

1994; Curry and Cushman, 1995a,b).

For instance, in an early work Bitsanis et al. (1990) examined the effect of shear on spherical, symmetric

molecules, conﬁned between planar, parallel walls that lacked atomic-scale roughness. Both Couette

(simple shear) and Poiseuille (pressure-driven) ﬂows were examined. The presence of ﬂow had no effect

on the density proﬁle of the liquids as they were identical with the equilibrium density proﬁles for both

types of ﬂow and all pore widths. Velocity proﬁles, deﬁned as the velocity of the liquid parallel to the

wall as a function of distance from the center of the pore, should be linear and parabolic for Couette

and Poiseulle ﬂow, respectively, for a homogeneous liquid. In the simulations of Bitsanis et al. (1990) the

velocity proﬁles for both types of ﬂow deviate from the shape expected for homogeneous ﬂow in the

FIGURE 11.21 Liquid density proﬁles in the direction normal to the surface of the pore walls as a function of pore

separation. The center of the pore is indicated by the single tick mark on the x-axis. Density proﬁles for all hydrocarbon

groups within the liquid molecules are represented with a solid line, for methyl groups 1 and 8 with open circles,

and for methylene groups 4 and 5 by crosses. The pore widths are 1.0 nm in (a), 1.2 nm in (b), 1.25 nm in (c),

1.4 nm in (d), 1.6 nm in (e), 1.8 nm in (f), 2.0 nm in (g), and 2.4 nm in (h). All liquid ﬁlms are at 297 K. (From

Wang, Y. et al. (1993), J. Phys. Chem. 97, 9013–9021. With permission.)

regions occupied by the two outermost ﬂuid monolayers. These outermost layers behaved like a ﬂuid of

very high viscosity.

The profoundly different nature of ﬂow in molecularly thin ﬁlms was further demonstrated by plotting

the effective viscosity vs. pore width (Figure 11.22). For a bulk material the viscosity was independent

of pore size. However, under both types of ﬂow, the viscosity increased slightly as the pore size decreased.

For ultrathin ﬁlms, the viscosity increased dramatically.

The effect of transverse order on conﬁned thin ﬁlms under shear, induced by atomic structure in the

walls, was also examined. Schoen et al. (1989) examined the behavior of argon LJ ﬁlms under shear,

conﬁned between fcc (100) walls, using both grand canonical Monte Carlo and microcanonical MD

simulations. At small wall separations of 2.0 σ, the pore accommodated two commensurate solid-like

layers when the walls were fully out of registry with one another. Shearing caused the stress to increase

linearly as the walls came closer into registry with one another. The number of atoms in the pore remained

essentially constant until a critical stress was reached, at which point continued shearing caused almost

an entire layer of ﬂuid to exit from the pore, thereby decreasing the stress. These results are in agreement

with experimental data, which indicates that solid surfaces slide past one another while separated by a

discrete number of layers. Further, a critical shear stress was required to initiate sliding under appropriate

conditions.

In contrast, with wall separations appropriate for an odd number of ﬂuid layers (3.10 σ and 4.90 σ)

the pore supported either three or ﬁve layers when the opposing surfaces were in registry. For the system

with three solid-like layers, the shear stress increased linearly when the system was sheared. As in the

previous case, this trend continued until a critical stress was achieved. Continued strain on the walls

resulted in the drainage of one of the solid layers.

The ﬂow of LJ liquids conﬁned between two solid walls was also examined by Thompson and Robbins

(1990b). In this case, the walls were composed of (001) planes of an fcc lattice. A number of wall and

ﬂuid properties, such as wall–ﬂuid interaction strength, ﬂuid density, and temperature, were examined.

The geometry of the simulations closely resembled the conﬁguration of an SFA (Figure 11.18b). Each

wall atom was attached to a lattice site with a spring to maintain a well-deﬁned solid structure with a

minimum number of solid atoms. The spring constant controls the thermal roughness of the wall and

its responsiveness to the ﬂuid, and it was adjusted so that the mean-square displacement about the lattice

FIGURE 11.22 Effective viscosity for Couette and Poiseuille ﬂows of an LJ liquid as a function of pore width. Pore

width is given in units of the LJ parameter σ (parameters used in this simulation were appropriate for argon). The

pore walls lack atomic-scale roughness. The viscosity of a homogeneous ﬂuid at the pore average density and that

of a bulk liquid are also shown. (From Bitsanis I. et al. (1990), J. Chem. Phys. 93, 3427–3431. With permission.)

sites was less than the Lindemann criterion for melting. The interactions between ﬂuid atoms and between

wall and ﬂuid atoms were modeled by different LJ potentials. Couette ﬂow was simulated by moving the

walls at a constant velocity in opposite directions along the x-axis (Figure 11.18). The heat generated by

the shearing of the liquid was dissipated using a Langevin thermostat. In most of the simulations

performed by Thompson and Robbins (1990b), the ﬂuid density and temperature were indicative of a

compressed liquid about 30% above its melting temperature, with 672 ﬂuid atoms, and either 192 or

352 wall atoms.

A number of interesting phenomena were observed in these simulations. First, the well-deﬁned lattice

structure of the solid walls induced both normal and parallel ordering in the adjacent liquid. The normal

ordering of the ﬁlms was apparent from the oscillations in the density proﬁles (Figure 11.19a to c).

Density oscillations in the liquid also induced oscillations in other microscopic quantities normal to the

walls such as the ﬂuid velocity in the ﬂow direction V

and the microscopic pressure–stress tensor P

(Figure 11.19d). The position of the peaks in P

corresponded to the peak positions in the liquid density,

while peak positions in dV

/dz are between the layers. These observations were contrary to the predictions

of the Navier–Stokes equations. However, it was determined that averaging dV

/dz and P

over length

scales that are larger than the molecular length σ produced smoothed quantities that satisﬁed the

Navier–Stokes equations (smooth line in Figure 11.19d).

While the ordering of the ﬂuid normal to the walls affected ﬂuid ﬂow, two-dimensional ordering of

the liquid parallel to the walls affected the ﬂow more signiﬁcantly. The velocity of the ﬂuid parallel to

the wall was examined as a function of distance from the wall for a number of wall–ﬂuid interaction

strengths and wall densities (Figure 11.23). Analysis of velocity proﬁles showed that ﬂow near solid

boundaries is strongly dependent on the strength of the wall–ﬂuid interaction and on wall density. For

instance, when the wall and ﬂuid densities were equal and wall–ﬂuid interactions strengths were small

(ε

/ε = 0.4), the velocity proﬁle was linear with a no-slip BC (Figure 11.23a). As the wall–ﬂuid interaction

strength increased, the magnitude of the liquid velocity in the layers nearest the wall increased; thus, the

FIGURE 11.23 The component of the liquid velocity in the

shearing direction V

vs. the position between the pore walls z.

The locations of the ﬁrst layers of solid atoms on each side of

the ﬂuid correspond to the vertical borders. Data points indicate

averages of V

within the layers and the solid lines represent

polynomial ﬁts to these data. The dashed line in (b) represents

the ﬂow expected from hydrodynamics with a no-slip BC. The

liquid velocity has been normalized by the velocity of the walls

and the pore width by the LJ length parameter σ. (From

Thompson, P. A. and Robbins, M. O. (1990), Phys. Rev. A 41,

6830–6837. With permission.)