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PHYSICAL REVIEW FLUIDS 4, 064201 (2019)

Universal molecular-kinetic scaling relation for slip

of a simple ﬂuid at a solid boundary

Gerald J. Wang and Nicolas G. Hadjiconstantinou*

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,

Massachusetts 02139, USA

(Received 24 August 2018; published 26 June 2019)

Using the observation that slip in simple ﬂuids at low and moderate shear rates is a ther-

mally activated process driven by the shear stress in the ﬂuid close to the solid boundary, we

develop a molecular-kinetic model for simple ﬂuid slip at solid boundaries. The proposed

model, which is in the form of a universal scaling relation that connects slip and shear rate,

reduces to the well-known Navier slip condition under low shear conditions, providing a

direct connection between molecular parameters and the slip length. Molecular-dynamics

simulations are in very good agreement with the predicted dependence of slip on system

parameters, including the temperature and ﬂuid-solid interaction strength. Connections

between our model and previous work, as well as simulation and experimental results,

are explored and discussed.

DOI: 10.1103/PhysRevFluids.4.064201

I. INTRODUCTION

Fluids under nanoscale conﬁnement can exhibit a number of remarkable transport properties,

including anomalous ﬂow rates [1–3], diffusion [4], and heat transfer [5]. Nanoﬂuidic engineering

exploits these phenomena for the development of novel materials for clean water [6] and energy

[7] among other applications. Modeling the dynamics of nanoconﬁned ﬂuids requires a detailed

understanding of the effect of the ﬂuid-solid interface on transport in these conditions [8–11].

Slip at the ﬂuid-solid interface is, perhaps, the most ubiquitous of these phenomena and has

received considerable attention (see, for example, Refs. [11–19]). In the case of dilute gas systems,

the functional form of the slip relation as well as slip coefﬁcients can be calculated via asymptotic

expansions of the Boltzmann equation [20–22]. In dense liquids such analytical treatments are not

possible; however, strong empirical evidence exists that the slip relation is of the same form as the

dilute case, namely,

us=β∂u

∂ηb

,(1)

which is referred to as the Navier slip condition, after Navier [23] who ﬁrst proposed it. In this

expression, usis the slip velocity (difference between the ﬂuid velocity at the boundary and the

boundary velocity), uis the ﬂow velocity in the direction parallel to the boundary, and ηis the wall

normal in the direction pointing into the ﬂuid; the subscript “b” denotes the boundary location.

Most research in the dense-ﬂuid arena has thus focused on investigating the properties of the

slip length β. Of particular note is the work of Thompson and Troian [11], which showed that

molecular-dynamics (MD) data for the slip length could be described well by an expression of the

form β=β0(1 −˙γ/˙γc)−1/2(where β0is the slip length at low shear rates, ˙γdenotes the shear

*Corresponding author: ngh@mit.edu

2469-990X/2019/4(6)/064201(10) 064201-1 ©2019 American Physical Society

GERALD J. WANG AND NICOLAS G. HADJICONSTANTINOU

rate, and ˙γcis a constant that depends on the ﬂuid as well as the ﬂuid-solid interaction details),

suggesting that the slip length obeys some form of critical dynamics. Other groups have made use

of Green-Kubo analyses to develop models that relate the slip length to the solid-liquid interaction

potential corrugations, at both the atomistic scale [14,24] and the roughness scale [16]. Several

authors [25–27] have proposed that slip exhibits many of the hallmarks of a thermally activated

process, at least for simple ﬂuids in contact with atomically smooth boundaries.

Despite this considerable progress, complete and predictive models of ﬂuid slip based on ab initio

(molecular) considerations have yet to be fully developed. The goal of the present work is to present

a physically motivated model for slip at the interface between a simple ﬂuid and a molecularly

smooth solid that is able to unify existing work and explain our, as well as previous, simulation

results.

II. FORMULATION

The model proposed here is based on the observation that ﬂuid slip on a molecularly smooth wall

at low and moderate shear rates can be modeled as a thermally activated process [26,27] and can

thus be quantitatively described using an extension of the Eyring theory of reaction rates [28,29].

For a discussion of this theory and its connection to transition state theory (TST), see Ref. [30].

Provided the conditions for an activated process are met, rate theory can be used to relate the drift

velocity of molecules under the inﬂuence of a driving force to the rate of hopping over the potential

barrier generated by nearby molecules. Eyring used such an approach to develop a theory for the

viscosities of dense ﬂuids [29], while Blake and coworkers pioneered the use of this concept to

describe slip and utilized it to model contact-line motion as a molecular-kinetic process [25]. The

model by Blake and Haynes (in particular, Ref. [25]) has been widely accepted in the contact-line

literature and has been found to be in good agreement with experimental data (see Refs. [31,32]for

discussions) as well as MD simulations of contact-line motion [33,34].

Following the exposition by Wyart and deGennes [35], the drift velocity, ud, can be written as

the difference between the forward and backward hopping rates ud=lj(κ+−κ−) with

κ±=τ−1

0exp −V∓1

2fdl2

j

kBT,

leading to

ud=2lj

τ0

exp −V

kBTsinh fdl2

j

2kBT,(2)

where ljis the jump length, τ0is the jump time scale, Vis the potential barrier associated with the

jump, kBis Boltzmann’s constant, and fdis the force per unit length acting on the ﬂuid molecules

causing this drift.

To make further progress, we make two observations. First, in the case of slip, the force on the

molecules at the ﬂuid-solid boundary responsible for the drift is the shear stress in the ﬂuid at this

location, μ(∂u/∂η)|b, where μdenotes the viscosity. Assuming that within a few atomic diameters

from the boundary an “inner” description exists in which molecular effects dominate, and noting

that the slip boundary condition is associated with the outer (Navier-Stokes) region description, we

interpret the above quantity as taken at the interface of the outer and inner regions, that is, in a region

where the Navier-Stokes description is still valid. In other words, μcorresponds to the bulk value

of the viscosity, while (∂u/∂η )|bcorresponds to the velocity gradient a few atomic diameters away

from the boundary where layering effects do not affect the ﬂow ﬁeld signiﬁcantly.

Our second observation is that, in its most general form, the potential Vrepresents the overall

potential landscape, and thus for a ﬂuid molecule at the ﬂuid-solid interface it includes both the ﬂuid-

solid (Vs) as well as ﬂuid-ﬂuid (Vf) interactions. Assuming additivity of the potential contributions,

which is certainly true for our simulations, exp(−V/kBT) can be factored into a term containing

064201-2

UNIVERSAL MOLECULAR-KINETIC SCALING RELATION …

the ﬂuid-solid interaction and a term containing the ﬂuid-ﬂuid interaction, which can be absorbed

into the timescale by deﬁning ˆτ0=τ0exp(Vf/kBT). As shown below, this allows us to explicitly

highlight the effect of ﬂuid-solid interaction.

Writing fdlj=μ−1

FL (∂u/∂η)|b, where FL denotes the areal density of ﬂuid molecules at the

ﬂuid-solid interface (number of molecules in the ﬁrst ﬂuid layer at the ﬂuid-wall interface per unit

interface area), we obtain the following expression for the slip velocity:

us=2lj

ˆτ0

exp −αε

kBTsinh μlj

2FLkBT˙γ,(3)

where, as a reminder, ˆτ0now includes the contribution of ﬂuid environment on the potential barrier.

In this expression, we have factored the overall ﬂuid-solid interaction energy Vsinto αε, where ε

is the energy scale for ﬂuid-solid interactions and αrepresents the potential energy of each ﬂuid

atom in the ﬁrst ﬂuid layer due to its interaction with all of the solid atoms, expressed in units

of ε. In other words, αis the scaled potential interaction energy of a single ﬂuid atom in the ﬁrst

ﬂuid layer summed (or, in the mean-ﬁeld sense, integrated) over all solid atoms. The properties and

characteristic values of the areal density, FL, and the related volumetric density of the ﬁrst ﬂuid

layer at the ﬂuid-solid interface, ρFL =FL/hFL (where hFL denotes the width of the ﬁrst layer),

have been extensively studied in a recent publication by the authors [36].

It immediately follows from (3) that in the small-shear-rate limit, ˙γ2FLkBT/(ljμ), Eq. (3)

linearizes to the Navier slip condition (1), with

β=μl2

j

FLkBTˆτ0

exp −αε

kBT.(4)

It is worth observing that, contrary to the continuum approach where βis a parameter whose

value needs to be supplied as part of the problem speciﬁcation, the molecular-kinetic approach

provides a direct connection between the slip length and the governing molecular parameters. As a

consequence, with very detailed micromechanical information, Eq. (4) could in principle be used to

directly predict the slip length from ﬁrst principles.

III. VALIDATION

To assess these ideas, we performed nonequilibrium MD simulations of plane-Couette ﬂow,

described in detail in the Appendix. Our results and subsequent discussion will be expressed in

terms of standard LJ nondimensional quantities [37], namely, σfor length, εffor energy, and

τ≡(mσ2/εf)1/2for time. In order to verify each of the dependences in Eq. (3), we measure the

slip velocity as we systematically vary the shear rate, the temperature, and the ﬂuid-solid interaction

strength.

We begin by studying the dependence on shear rate. Figure 1shows a comparison between MD

simulation results and the prediction of Eq. (3) scaled in the form

us=u0sinh( ˙γ/˙γ0).(5)

To test this scaling, we generated data sets in the above-described geometry in which the shear

rate was varied, while all other parameters were held constant. To augment these data sets, we also

collected all rigid-wall MD simulation data from Refs. [11] and [38] and scaled those according to

(5). In plotting the scaled data, the constants u0≡2lj

ˆτ0exp (−αε

kBT) and ˙γ0≡2FLkBT

μljwere determined

for each data set by means of a nonlinear least-squares ﬁt to (5). For in-house MD simulations

(shown in blue), the vertical size of the symbols reﬂects the characteristic scale of uncertainty in

the corresponding slip velocity measurements. In particular, the symbol height is equal to the width

of the 95% conﬁdence interval on the slip velocity, as determined via a linear ﬁt to the velocity

proﬁle in the bulk region (as described in the Appendix). The ﬁgure shows that Eq. (5)isableto

accurately describe all data sets, including those reported by Thompson and Troian, which were

064201-3

GERALD J. WANG AND NICOLAS G. HADJICONSTANTINOU

FIG. 1. Scaled slip velocity as a function of scaled shear as measured in MD simulations. The scaling

predicted by Eq. (3) is shown as the dashed green line (slip model). MD data transcribed from [11] (TT ’97)

and [38] (MHPL ’08) and scaled according to Eq. (3) are shown in green and pink; in-house MD results are

showninblue.

shown to be ﬁtted well by the expression β=β0(1 −˙γ/˙γc)−1/2[11]. In other words, although both

(5) and the model by Thompson and Troian ﬁt the data of Fig. 1with comparable accuracy in view

of the uncertainty in the data, compared to the latter, expression (5) has the beneﬁt of being more

physically motivated, due to both its clear connection to a physical model of slip and the fact that it

predicts ﬁnite slip at all ﬁnite shear rates. In fact, (5) is also able to describe well the slip velocities

measured in a wide range [39–41] of experiments (see Fig. 2) involving aqueous solutions, alkanes,

FIG. 2. Scaled slip velocity as a function of scaled shear for experiments performed via atomic force

microscopy [39] (CNW ’01), surface-force apparatus [40] (ZG ’01), and near-ﬁeld laser velocimetry [41] (LL

’03). The scaling predicted by Eq. (3) is shown as the dashed green line (slip model).

064201-4

UNIVERSAL MOLECULAR-KINETIC SCALING RELATION …

TABLE I. Parameters for characteristic length and time scales in (3), for each of the densities in Fig. 1.

ρave lj[Å] ˆτ0exp ( αε

kBT)[ns]

0.6 1.6±0.29 0.034 ±0.009

1.0 2.4±0.35 0.92 ±0.51

and polymeric ﬂuids. In both ﬁgures, we are able to observe a regime of shear rates consistent with

the Navier slip relation (for ˙γ/˙γ01, us∝˙γ), beyond which the slip velocity rises dramatically

with shear rate.

Using values of the viscosity μfrom Ref. [42] and values of the areal density of interfacial

ﬂuid FL from Ref. [36], we are able to infer the values for the molecular-kinetic parameters in (3),

namely, ljand ˆτ0exp ( αε

kBT) for our in-house MD data. These values, along with their associated 95%

conﬁdence interval, are presented in Table I. We observe that both parameters are of molecular scale

as expected. In fact, ljlies between the graphene interatomic spacing (a=1.42 Å) and the ﬂuid-

ﬂuid characteristic spacing (σ=3.15 Å), demonstrating consistency with the proposed molecular

mechanism. Here we note that the slight increase of ljwith ρave suggests that the effect of ﬂuid-

ﬂuid interaction becomes more important as the density is increased, as expected. Along the same

lines, we also note the sensitivity of the product ˆτ0exp ( αε

kBT)toρave; this can be understood by

noting that Vfis expected to increase with ρave , again demonstrating consistency with the proposed

molecular mechanism. Further analysis of the complete term ˆτ0exp ( αε

kBT), including development

of approaches for separating the contributions of its constituents, will be undertaken in the future.

We now investigate the other major factors affecting slip, namely, the temperature and the

ﬂuid-solid interaction strength. Having veriﬁed the nonlinear behavior with shear rate in the above

section, the following comparisons will be performed for low shear rates for which an explicit

expression for the slip length exists [see (4)].

Figure 3shows the temperature dependence of the slip length at a ﬁxed ﬂuid-solid interaction

strength ε=1 and boundary speed uw=0.25, for two ﬂuid densities. We ﬁnd that the slip length

FIG. 3. Slip length as a function of temperature for high- and low-density ﬂuids (uw=0.25), where the

change in nondimensional temperature is effected by changes in the dimensional temperature (εis held constant

at 1). For both densities, the results from MD simulation show strong agreement with the dependence in (6)

(slip model).

064201-5

GERALD J. WANG AND NICOLAS G. HADJICONSTANTINOU

FIG. 4. Slip length as a function of ﬂuid-solid interaction strength for high- and low-density ﬂuids (uw=

0.125). For both densities, the results from MD simulation show strong agreement with the exponential decay

given by (7) (slip model).

is ﬁtted well (R20.89) by the form

β=c1exp c2

T,(6)

where c1and c2are ﬁtting constants. This form represents the dominant temperature dependence in

(4) arising from term exp(−V/kBT). Additional contributions to the temperature dependence can be

accounted for by noting that (a) for a Lennard-Jones ﬂuid, μ∼Tζwith ζ≈1 when ρ0.6 and

T2[43] and (b) the leading-order temperature dependence of FL was shown in Ref. [36]tobe

of the form FL ∼a+bWa, where Wa ∝ε/(kBT) is the Wall number introduced in Ref. [36] and

aand bare density-dependent constants introduced and discussed in Sec. II C of the same reference.

Modifying expression (6)toβ=c1exp(c2/T)Tγ(a+b/T)−1, where Tγrepresents any residual

temperature dependence (e.g., τ0or due to the difference between ζand 1) and using values of a

and bas calculated from independent MD simulations in Ref. [36], has a negligible effect on the ﬁt

quality (R2value increase of 1% or less).

Figure 4shows the dependence of the slip length on εat a ﬁxed temperature T=5 and boundary

speed uw=0.125, for two ﬂuid densities. We ﬁnd that the form

β=c3exp(−c4ε),(7)

where c3and c4are ﬁtting constants, results in a very strong ﬁt (R20.91). This form represents

the dominant dependence of βon εin (4). Including the dependence of FL on εby modifying

expression (7)toβ=c3exp(−c4ε)(a+bε)−1and using values of aand b as calculated in

Ref. [36] has a negligible effect on the ﬁt quality (R2value increases from 0.971 to 0.973). We

also note that for the set of simulations depicted in Fig. 4the slip length is essentially independent

of the liquid density (in fact, both data sets are jointly well ﬁtted by the same exponential decay).

This is consistent with the results of Fig. 3, which shows that the slip length is nearly independent

of density for T5. As a ﬁnal note on the consistency of the two ﬁts [(6) and (7)], we point out

that at the condition T=5 and ε=1, both predict slip length values of approximately 16, for both

ﬂuid densities.

064201-6

UNIVERSAL MOLECULAR-KINETIC SCALING RELATION …

IV. CONNECTION TO OTHER STUDIES OF SLIP

The present model is, in general, in good agreement with earlier theoretical and experimental

work on liquid slip. For example, Eq. (3) predicts that at ﬁxed temperature, in the small-shear-rate

limit, slip is linear in the bulk viscosity. This is in good agreement with the Green-Kubo analysis by

Barrat and Bocquet [14,24], who found the same scaling; this observation also agrees qualitatively

with the predictions of the variable-density Frenkel-Kontorova model [15]. The strength of this

result is even clearer after careful comparison with experiments that probe the effect of varying both

the viscosity and the shear rate [39] across the transition into the nonlinear regime. The data of Craig

et al., included in Fig. 2, show both a transition to the nonlinear regime as the shear rate is increased

at ﬁxed viscosity, but also that the controlling factor for this transition is the product of viscosity

and shear rate. Their data show quantitative agreement (within 14% error) with the predictions of

(3)(seeFig.2).

Equation (3) is also in good qualitative agreement with earlier work focused on the relationship

between slip and wettability. Both MD simulations [11,44] and experiments [45,46] have found that

slip tends to increase as the ﬂuid becomes less wetting (as the energy scale εof ﬂuid-solid interaction

decreases). More speciﬁcally, the MD simulations in Ref. [44] strongly suggest that dβ/dε<0 and

d2β/dε2>0, i.e., slip decays as a convex function of ε, which is reﬂected by Eq. (3). We also

note that under some assumptions, Green-Kubo theory [14] predicts β∝ε−2(for ﬁxed ﬁrst layer

density and ﬁrst layer structure factor), which is in qualitative agreement with the above ﬁndings;

this behavior was also observed in MD simulations of patterned wettability [47] in the cases where

the slip is determined by the interaction between the ﬁrst ﬂuid layer and the wall. Here we also note

that in the limit ε→0, where the barrier height associated with the ﬂuid-solid interaction becomes

small, we expect (3) to no longer hold.

As explained above, the presence of FL in Eq. (3) provides a direct connection between slip and

ﬂuid layering at the ﬂuid-solid interface; this connection was ﬁrst studied by Barrat and Bocquet

[14] in atomically smooth settings using Green-Kubo theory. The latter study has shown that, under

certain assumptions, and under ﬁxed temperature, ﬂuid-solid interaction and ﬁrst layer structure

factor as well as diffusion coefﬁcient in the direction parallel to the ﬂuid-solid interface in the ﬁrst

ﬂuid layer, β∝1/ρFL. Careful MD simulations [18] have veriﬁed that the slip length and ρFL are

inversely related, but with a slightly different exponent, namely, β∼ρ−1.44

FL . These results are in

general agreement with the prediction of Eq. (4); the latter can be seen more clearly by noting that

FL =ρFLhFL and that the dependence of hFL on density is weak [36]. It also needs to be noted

that the remaining independent variables appearing in our expression are not exactly the same as the

expression of Barrat and Bocquet. The agreement between Green-Kubo theory and MD simulations

was also studied in the presence of physical surface corrugations (wall roughness) in Ref. [16]; this

is an important consideration that merits further examination from the rate-process perspective in

the future.

V. CONCLUSIONS AND OUTLOOK

We have proposed and validated a general scaling relation describing slip of simple ﬂuids at

smooth solid boundaries. The model builds upon the observation that slip is a thermally activated

process, proposed by others [25] and more ﬁrmly established by recent careful studies [26]. The

proposed scaling relation is found to be in excellent agreement with MD simulations as well

as experimental data; the latter even extends to moderately complex ﬂuids. In other words, the

proposed model provides further evidence that slip at low and moderate shear rates can be modeled

as a rate process but also provides an explicit expression for predicting slip in terms of molecular

system parameters. Although the power-law relation proposed in Ref. [11] also exhibits good

agreement with MD simulations, Eq. (3) has the advantage of being associated with a clear physical

model of the slip process and does not suffer from a slip-length divergence at ﬁnite shear rates.

064201-7

GERALD J. WANG AND NICOLAS G. HADJICONSTANTINOU

TheworkbyMartiniet al. [38] suggests that in the high rate limit, slip is no longer a rate

process. In this work we have limited our MD simulations to low and moderate shear rates given

by ˙γτ 0.07, such that, in addition to low-to-moderate shear rates, we ensure μ= μ(˙γ). It is

worth emphasizing that, from a practical perspective, this low to moderate shear-rate regime covers

virtually all nanoﬂuidic engineering applications (for channels of nanoscale dimensions and typical

ﬂuids, this condition corresponds to ﬂow velocities O(102)m/s). On the other hand, we note that

at high shear rates Eq. (3) predicts a plateau for the slip length (shear thinning in the large- ˙γlimit

typically results in a behavior of the type μ∝˙γ−1for simple ﬂuids [48]), in qualitative agreement

with the predictions of the Frenkel-Kontorova model and MD simulations [38]. A quantitative

investigation of the high-shear-rate limit will be the subject of future work.

ACKNOWLEDGMENTS

The authors are grateful for support from the DOE CSGF (Contract No. DE-FG02-97ER25308)

as well as the Center for Nanoscale Materials, a U.S. Department of Energy, Ofﬁce of Science,

Ofﬁce of Basic Energy Sciences User Facility (Contract No. DE-AC02-06CH11357).

APPENDIX: MOLECULAR-DYNAMICS SIMULATIONS

We performed nonequilibrium MD simulations using the LAMMPS code [49]. Our system

consisted of a Lennard-Jones (LJ) ﬂuid [37] of density ρave,atomicmass16g/mol, length scale

σ=3.15 Å, and energy scale εf=0.15 kcal/mol, in a plane-Couette setup of channel width 50 Å,

with periodic boundary conditions in all directions. We veriﬁed that our conclusions are not affected

by channel width, by running a small subset of simulations in channels of width 25 Å and 100 Å. In

this Appendix, we describe our setup in detail, nondimensionalizing using the dimensions provided

above.

The ﬂuid is conﬁned between two rigid sheets of graphene at ﬁxed zcoordinate, which move

in opposite directions at ﬁxed velocity uw. In each simulation, after an equilibration period of 3 ×

103with a Nosé-Hoover thermostat [50] at a temperature of 5 (unless otherwise speciﬁed), ﬂuid

velocities are averaged over a time period of 6 ×103to obtain a velocity proﬁle as a function of the

zcoordinate; to reduce variance in low-signal simulations [51], the averaging period was extended

to 3 ×104if us0.01. The use of a thermostat is necessary in these calculations in order to regulate

temperature in the presence of viscous heating. For a small subset of simulations, we veriﬁed that

an alternative choice of the thermostat (in particular a Berendsen thermostat [52]) does not affect

our conclusions. We also veriﬁed that the results obtained using no thermostat (simulations in the

microcanonical ensemble) agree with the results from both thermostats, provided that the shear rate

is low (and so viscous heating is negligible).

All simulations were performed in the regime ˙γτ < 0.07, where it is well known that simple

ﬂuids have shear-rate-independent viscosity [11,19]. We determine the slip velocity by ﬁtting

a line to the ﬂuid velocity proﬁle away from the solid boundaries (which can induce strong

inhomogeneities in the ﬂuid density [8,36]) and extrapolating to the locations of the walls. In our

simulations, we ﬁt the velocity proﬁle over the central region of the channel, at least three distance

units away from the walls.

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