Bhushan B. Nanotribology and Nanomechanics: An Introduction

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514 Marina Ruths and Jacob N. Israelachvili

351. J. Peachey, J. Van Alsten, S. Granick: Design of an apparatus to measure the shear

response of ultrathin liquid ﬁlms, Rev. Sci. Instrum. 62, 463–473 (1991)

352. E. Rabinowicz: The intrinsic variables aﬀecting the stick–slip process, Proc. Phys. Soc.

71, 668–675 (1958)

353. T. Baumberger, F. Heslot, B. Perrin: Crossover from creep to inertial motion in friction

dynamics, Nature 367, 544–546 (1994)

354. F. Heslot, T. Baumberger, B. Perrin, B. Caroli, C. Caroli: Creep, stick–slip, and dry-

friction dynamics: Experiments and a heuristic model, Phys. Rev. E 49, 4973–4988

(1994)

355. J. Sampson, F. Morgan, D. Reed, M. Muskat: Friction behavior during the slip portion

of the stick–slip process, J. Appl. Phys. 14, 689–700 (1943)

356. F. Heymann, E. Rabinowicz, B. Rightmire: Friction apparatus for very low-speed slid-

ing studies, Rev. Sci. Instrum. 26, 56–58 (1954)

357. J.H. Dieterich: Time-dependent friction and the mechanics of stick–slip, Pure Appl.

Geophys. 116, 790–806 (1978)

358. J.H. Dieterich: Modeling of rock friction.1. Experimental results and constitutive equa-

tions, J. Geophys. Res. 84, 2162–2168 (1979)

359. G.A. Tomlinson: A molecular theory of friction, Phil. Mag. 7, 905–939 (1929)

360. J.M. Carlson, J.S. Langer: Mechanical model of an earthquake fault, Phys. Rev. A 40,

6470–6484 (1989)

361. B.N.J. Persson: Theory of friction: The role of elasticity in boundary lubrication, Phys.

Rev. B 50, 4771–4786 (1994)

362. H. Yoshizawa, J. Israelachvili: Fundamental mechanisms of interfacial friction.

2: Stick–slip friction of spherical and chain molecules, J. Phys. Chem. 97, 11300–

11313 (1993)

363. S. Nasuno, A. Kudrolli, J.P. Gollub: Friction in granular layers: Hysteresis and precur-

sors,Phys.Rev.Lett.79, 949–952 (1997)

364. M.O. Robbins, P.A. Thompson: Critical velocity of stick–slip motion, Science 253,

916 (1991)

365. P. Bordarier, M. Schoen, A. Fuchs: Stick–slip phase transitions in conﬁned solidlike

ﬁlms from an equilibrium perspective, Phys. Rev. E 57, 1621–1635 (1998)

366. J.M. Carlson, A.A. Batista: Constitutive relation for the friction between lubricated

surfaces, Phys. Rev. E 53, 4153–4165 (1996)

367. A.D. Berman, W.A. Ducker, J.N. Israelachvili: Origin and characterization of diﬀerent

stick–slip friction mechanisms, Langmuir 12, 4559–4563 (1996)

368. A.D. Berman, W.A. Ducker, J.N. Israelachvili: Experimental and theoretical investi-

gations of stick–slip friction mechanisms, NATO ASI Ser. E: Appl. Sci. 311, 51–67

(1996)

369. KG. McLaren, D. Tabor: Viscoelastic properties and the friction of solids. Friction of

polymers and inﬂuence of speed and temperature, Nature 197, 856–858 (1963)

370. K.A. Grosch: Viscoelastic properties and friction of solids. Relation between friction

and viscoelastic properties of rubber, Nature 197, 858–859 (1963)

371. L. Bureau, T. Baumberger, C.Caroli: Shear response of a frictional interface to a normal

load modulation, Phys. Rev. E 62, 6810–6820 (2000)

372. J.P. Gao, W.D. Luedtke, U. Landman: Structure and solvation forces in conﬁned ﬁlms:

Linear and branched alkanes, J. Chem. Phys. 106,

4309–4318 (1997)

373. J. Warnock, D.D. Awschalom, M.W. Shafer: Orientational behavior of molecular liq-

uids in restricted geometries, Phys. Rev. B 34, 475–478 (1986)

374. D.D. Awschalom, J. Warnock: Supercooled liquids and solids in porous glass, Phys.

Rev. B 35, 6779–6785 (1987)

9 Surface Forces and Nanorheology of Molecularly Thin Films 515

375. M. Hirano, K. Shinjo, R. Kaneko, Y. Murata: Anisotropy of frictional forces in mus-

covite mica, Phys. Rev. Lett. 67, 2642–2645 (1991)

376. T. Gyalog, H. Thomas: Friction between atomically ﬂat surfaces, Europhys. Lett. 37,

195–200 (1997)

377. B.N.J. Persson, F. Bucher, B. Chiaia: Elastic contact between randomly rough surfaces:

comparison of theory with numerical results, Phys. Rev. B 65, 184106/1–7 (2002)

378. S. Hyun, L. Pei, J.-F. Molinari, M.O. Robbins: Finite-element analysis of contact be-

tween elastic self-aﬃne surfaces, Phys. Rev. E 70, 026117/1–12 (2004)

379. J. Israelachvili, N. Maeda, K.J. Rosenberg, M. Akbulut: Eﬀects of sub-ångstrom (pico-

scale) structure of surfaces on adhesion, friction and bulk mechanical properties, J.

Mater. Res. 20, 1952–1972 (2005)

380. T.R. Thomas: Rough Surfaces, 2nd edn. (Imperial College Press, London 1999)

Interfacial Forces and Spectroscopic Study

of Conﬁned Fluids

Y. Elaine Zhu, Ashis Mukhopadhyay,and Steve Granick

Summary. In this chapter we discuss three speciﬁc issues which are relevant for liquids in

intimate contact with solid surfaces. (1) Studies of the hydrodynamic ﬂow ofsimple and com-

plex ﬂuids within ultra-narrow channels show the eﬀects of ﬂow rate, surface roughness and

ﬂuid–surface interaction on the determination of the boundary condition. We draw attention

to the importance of the microscopic particulars to the discovery of what boundary condition

is appropriate for solving continuum equations and the potential to capitalize on slip at the

wall for purposes of materials engineering. (2) We address the long-standing question of the

structure of aqueous ﬁlms near a hydrophobic surface. When water was conﬁned between ad-

joining hydrophobic and hydrophilic surfaces (a Janus interface), giant ﬂuctuations in shear

responses were observed, which implies some kind of ﬂickering, ﬂuctuating complex at the

water–hydrophobic interface. (3) Finally we discuss recent experiments that augment fric-

tion studies by measurement of diﬀusion, using ﬂuorescence correlation spectroscopy (FCS).

Here spatially resolved measurements showed that translation diﬀusion slows exponentially

with increasing mechanical pressure from the edges of a Hertzian contact toward the center,

accompanied by increasingly heterogeneous dynamical responses. This dynamical probe of

how liquids order in molecularly thin ﬁlms fails to support the hypothesis that shear produces

a melting transition.

10.1 Introduction

Conﬁnementof ﬂuids,rangingfromwater,hydrocarbonoil,polymermelts andsolu-

tions, to DNA, protein and other bio-macromolecules, can strongly aﬀect the struc-

tures and dynamics of the ﬂuid molecules, particularly when the thickness of ﬂuid

ﬁlms becomes comparable to the length scale of the molecular dimension. Phase

transitions, such as solidiﬁcation or glass transition, can be induced, and conﬁne-

ment can not only result in structural changes in terms of molecular orientation

and packing, but changes of dynamical processes, such as molecular translational

as well as rotational diﬀusion. These questions about conﬁned ﬂuid structure and

dynamics involve deep scientiﬁc puzzles. On the applied side, they also pertain

directly to the understanding of many fundamental issues in nature and technol-

ogy, including various applications in adhesion, colloidal stabilization, lubrication,

518 Y. Elaine Zhu et al.

micro/nanoﬂuidics, and micro/nano-electromechanical systems (MEMS/NEMS)

devices, among others.

The techniques that have been used to study conﬁned ﬂuids are mainly surface

forces apparatus (SFA) [1–4] and recently atomic force microscopy (AFM) and

its derivatives [5, 6]. SFA, which was originally designed to directly measure the

van der Waals and Derjaguin–Landau–Verwey–Overbeek(DLVO) forces in simple

liquids and in colloidal systems, was later modiﬁed by several groups to study the

frictional forces of molecularly thin ﬁlms conﬁned between rigid surfaces [2–4].

AFM is an alternative method to study surface force, friction, adhesion and lubrica-

tion of thin ﬁlm. With these two techniques many new phenomena about conﬁned

ﬂuids, such as layering structure, solidiﬁcation, stick–slip motion, etc. have been

experimentally observed and interpreted.

The previous Chap. 9 reviewed the advances in the studies of surface forces,

adhesion and shear interaction of conﬁned liquids. In this chapter, we review recent

studieson some speciﬁcinterfacialforces of thin liquidﬁlms by using modiﬁedSFA

devices and the results from an integrated experimental platform, which enables di-

rect ﬂuorescence-based observation of the dynamics of individual molecules under

conﬁnement.

This chapter is organized as follows. This introductory section is followed by

a discussion of the validity of the no-slip boundary for ﬂuid molecules moving past

a solid surface, determined by measuring the hydrodynamicforce in ﬂow geometry.

A speciﬁc interfacial force – the long-range hydrophobic interaction – and its re-

lated hydrophobic eﬀect on the behavior of water molecules near a hydrophobic

interface are discussed in Sect. 10.3. In Sects. 10.4–10.6, recent developments in

combining ultrafast spectroscopy with the SFA to investigate the dynamic behavior

of individual conﬁned molecules, instead of their ensemble-averaged behavior, are

reviewed.

10.2 Hydrodynamic Force of Fluids Flowing in Micro-

to Nanoﬂuidics: A Question About No-Slip Boundary Condition

Viscous ﬂow is familiar and useful, yet the underlying physics is surprisingly subtle

and complex. A tenet of textbook continuum ﬂuid dynamics is the no-slip boundary

condition, which means that ﬂuid molecules immediately adjacent to the surface of

a solid move with exactly the same velocity as that solid. It stands at the bedrock of

our understandingof the ﬂow of simple low-viscosity ﬂuids and comprisesa spring-

board for much sophisticated calculation. While it is true that at some level of detail

this continuum description must fail and demand description at the molecular level,

it is tremendously successful as the basis for continuum-based calculations. This

section is adapted from discussions in some previous primary accounts [7–9].

As expressed in a prominent ﬂuid dynamics textbook [10]:

“In other words there is no relative motion between the ﬂuid and the solid. This

fact may seem surprising but it is undoubtedlytrue. No matter how smooth the solid

10 Interfacial Forces and Spectroscopic Study of Conﬁned Fluids 519

surface or how small the viscosity of the ﬂuid, the particles immediately adjacent to

the surface do not moverelative to it.It is perhapsnot withoutinterestthat Newton’s

term for viscosity was ‘defectus lubricitatis’ – ‘lack of slipperiness’. Even for a ﬂuid

that does not ‘wet’ the surface this rule is not violated.”

Feynman noted in his lectures that the no-slip boundary condition explains why

large particles are easy to remove by blowing past a surface, but small particles are

not.

However, for many years it was observed that the no-slip boundary condition

is not intuitively obviously and has been controversial for centuries. A pantheon of

great scientists – among them, Bernoulli, Coulomb, Navier, Poisson, Stokes, Tay-

lor, Debye, de Gennes – has worried about it. The compelling rational arguments –

for and against – were divided by the pragmatic observation that predictions agree

with experiments. The possibility of slip was discussed until recently,in mainstream

literature, only in the context of the ﬂow of polymer melts [11, 12], though over

the years persistent doubts were expressed [13]. No-slip contrasts with slip char-

acteristic of highly viscous polymers [11,12], monolayers of gas condensed on vi-

brated solids [14], superﬂuidhelium [15], movingcontact lines of liquid droplets on

solids [16], and kinetic friction of liquid ﬁlms less than 5–10 molecular dimensions

thick [17,18]. However,experimental capability and technical needs have changed –

especially so with the emerging interest in microﬂuidics and microelectromechani-

cal system (MEMS)-based devices.

The situation has changed but the enormous and enduring success of the no-slip

assumption for modeling must be emphasized. It works beautifully provided that

certain assumptions are met: a single-component ﬂuid, a wetted surface, and low

levels of shear stress. Then careful experiments imply that the ﬂuid comes to rest

within 1–2 molecular diameters of the surface [19–21]. But the necessary assump-

tions are more restrictive, and their applicability is more susceptible to intentional

control than is widely appreciated. Generally, one may argue from the fact that ﬂuid

molecules are (“obviously”) stuck to walls by intermolecular forces. The traditional

explanation is that, since most surfaces are rough, the viscous dissipation as ﬂuid

ﬂowspastsurface irregularitiesbringsit to rest, regardlessof howweakly or strongly

molecules are attracted to the surface [7,8,22–24]. This has been challenged by ac-

cumulating evidence that, if molecularly smooth surfaces are wet only partially,

hydrodynamicmodels work better when one uses instead partial slip boundarycon-

ditions [13,21,25–30]. Then the main issue would be whether the ﬂuid molecules

attract the surface or the ﬂuid more strongly [9,13,27–34].

10.2.1 How to Quantify the Amount of Slip

To knowwhat boundary conditionis appropriatein solving continuum equationsre-

quires inquiry into microscopic particulars. Attention is drawn to unresolved topics

of investigation and to the potential to purposively capitalize on slip at the wall for

purposes of materials engineering.

The formal idea of a slip length is common. Slip signiﬁes that, in the continuum

model of ﬂow, the ﬂuid velocity at the surface is ﬁnite (the slip velocity, υ

)and

520 Y. Elaine Zhu et al.

True slip Apparent slip

Low

viscosity

zone

Fig. 10.1. Schematic illustration of the distinction between true slip (left) and apparent slip

(right) in oscillatory ﬂow, although for both cases the velocity of the moving ﬂuid extrapo-

lates to zero at a notional distance inside the wall and is ﬁnite where it crosses the wall. For

true slip, this is literally so. It may also happen that a low-viscosity component in the ﬂuid

facilitates ﬂow because it segregates near the surface. The velocity gradient is then larger

nearest the surface because the viscosity is smaller. When speciﬁc real systems are inves-

tigated, structural and chemical materials analysis at the microscopic scale are needed to

distinguish between these possibilities. After [7] with permission

increases linearly with distance normal to it. The slip velocity is assumed to be

proportional to the shear stress at the surface, σ

in the relation:

ηυ

= bσ

, (10.1)

where η is the viscosity and b, the slip length, is the notional distance inside the

surface at which the velocity equals zero, if thevelocityproﬁle is extrapolated inside

the surface until it reaches zero. In much of the literature the slip length has been

assumed to be a constant that characterizes the material response of a given ﬂuid–

surface pair, but evidence of additional dependence on velocity is discussed below.

It is unreasonable to expect this continuum description to yield microscopic in-

formation. One example of this was already given – the appearance of no-slip con-

ditions when the microscopic reason is that ﬂow irregularities pin the ﬂuid to the

wall [7,8,22–24]. Another example is apparent slip when a low-viscosity compo-

nent in the ﬂuid facilitates ﬂow because it segregates near the surface [35,36]. When

conventional continuum mechanics contends with situations that are more complex

than the model allows, one should resist the temptation to interpret literally the pa-

rameters in which the continuummechanics model is couched. These examplesem-

phasize instances where the notionsof stick and slip are numericalconveniencesnot

to be interpreted literally in terms of molecular mechanism. The appearance of slip

owingto surface segregationof a low-viscositycomponent is illustrated in Fig. 10.1.

10.2.2 The Mechanisms that Control Slip in Low-Viscosity Fluids

Partial slip of so-called Newtonian ﬂuids such as alkanes and water is predicted by

an increasing numberof computer simulations[25–27,37–39] and, in the laboratory

when forces are measured, systematic deviationsfrom the predictions basedon stick

10 Interfacial Forces and Spectroscopic Study of Conﬁned Fluids 521

are found [7–9,28–34,40,41]. Some sense of urgency comes from potential practi-

cal applications. Typical magnitudes of the slip length reported in the literature are

submicron, so small that the practical consequences of slip would be minimal for

ﬂow in channels whose size is macroscopic. But if the channel size is very small,

there are potential ramiﬁcations for microﬂuidics.

The simulations must be believed because they are buttressed by direct meas-

urements. In the past, all laboratory reports of slip were based on comparing me-

chanical measurements of force to ﬂuid mechanics models and hence were indirect

inferences. Recently, optical methods were introduced to measure ﬂuid velocity di-

rectly. For example, Léger andcoworkers photobleachedtracer ﬂuorescent dyes and

from the time rate of ﬂuorescence recovery, measured in attenuated total reﬂection

in order to focus on the region within an optical wavelength of the surface, the ve-

locity of ﬂow near the surface was inferred [28]. They reported slip of hexadecane

near an oleophobic surface provided that it was smooth, but not when it was rough.

Tretheway and Meinhart used laser particle image velocimetry of tracer latex par-

ticles to infer the velocity of water ﬂow in microchannels [34]. They reported slip

when the surface was hydrophobic but stick when it was hydrophilic. Callahan and

coworkers demonstrated the feasibility of using nuclear magnetic resonance (NMR)

velocityimaging, thoughthis method hasbeen usedto date onlyfor multicomponent

ﬂuids [42].

An important hint about the mechanism comes from the repeated observation

that the amount of slip depends on the ﬂow rate, in measurements using not only

the surface forces apparatus (SFA) [8, 9,31,32] but also atomic force microscopy

(AFM) [29, 41]. The main idea of all of these experiments is that two solids of

mean radius of curvature R, at spacing D, experience a hydrodynamic force, F

they approach one another (or retreat fromone another)in a liquid medium, thereby

squeezing ﬂuid out of (or into) the intervening gap. This force F

is proportionalto

the rate at which the spacing changes, dD/dt (where t denotes time), is proportional

to the viscosity, η (assumed to be constant), and is inversely proportional to D.The

no-slip boundary condition combined with the Navier–Stokes equations gives to

ﬁrst order the following expression, known as the Reynolds equation:

= f

6πR

. (10.2)

Higher-ordersolutionsof the Navier–Stokesequations conﬁrmthat the lowest-order

term is enough. The deviation of the dimensionless number f

∗

from unity quantiﬁes

the deviation from the classical no-slip prediction. The classical prediction is analo-

gous when the surface spacing is vibrated. In that case a sinusoidal oscillatory drive

generates an oscillatory hydrodynamic force whose peak we denote as F

H,peak

.The

peak velocity of vibration is υ

peak

= d ·ω,whered is the vibration amplitude and

ω the angular frequency of vibration. Studies show that, when the frequency and

amplitude of oscillatory ﬂow are varied, results depend on their product, υ

peak

and

the deviations from (10.2) depend on υ

peak

/D. This ratio, the ﬂow rate, is the ratio

suggested by the form of (10.2).

522 Y. Elaine Zhu et al.

Without necessarily assigning physical meaning to this quantity, it can be used

as an alternativeexpressionof the samedata. Mathematicalmanipulation[43] shows

that f

∗

and the slip length, b are related as:

∗

= 2











−1



. (10.3)

10.2.3 Experimental

The slippery question was studiedexperimentallyby testing thelimits of ideasabout

interfacial force and surface roughness. Figure 10.2 sketches the experimental strat-

egyand showsAFM (atomicforce microscopy)imagesof some ofthe surfacesstud-

ied. Three strategies were used to vary surface roughness systematically. The ﬁrst

was based on using collapsed polymers. Narrow-distribution samples of diblock

copolymers of polystyrene (M

= 55,400) and polyvinylpyridine (M

= 28,200),

PS/PVP, were allowed to adsorb for a limited time from dilute (5×10

µg/ml)

toluene solution onto freshly cleaved muscovite mica. They appeared to aggregate

during the drying process. The remaining bare regions of mica were then coated

with an organic monolayer of condensed octadecyltriethoxysilane (OTE) [44]. The

contact angles against water and tetradecane were stable in time, which shows that

the liquids did not penetrate them. AFM images in Fig. 10.2 quantify the roughness.

To produce larger roughness values, mica was coated with condensed octadecyl-

trichlorosilane (OTS) after ﬁrst saturating the cyclohexane deposition solution with

O to encourage partial polymerization in solution before surface attachment. In

the third method, self-assembled monolayers of alkane thiols were formed on silver

whose roughness was controlled by a direct-current (DC) bias applied during sput-

Scheme:

Flow

Case a) b) c)

Fig. 10.2a–c. The scheme of ﬂow over a rough surface is shown schematically in the top

portion. In the bottom panel, AFM images areshown of the following case: (a) self-assembled

OTS layers; (b)PS/PVP-OTE layers (surface coverage ≈ 80%); (c)PS/PVP-OTE layers

(surface coverage ≈ 20%). Each AFM image concerns an area 3 µm×3 µm. After [8] with

permission

10 Interfacial Forces and Spectroscopic Study of Conﬁned Fluids 523

ter deposition. The silver was coated with octadecanethiol deposited from ethanol

solution and placed in opposition with a molecularly smooth OTE surface.

Accordingto (10.2), thestick prediction (f

∗

= 1) correspondsto a horizontalline

and one observes in Fig. 10.3 that deviationsfrom this prediction decreased system-

atically as the surface roughness increased. In addition, deviations from the predic-

tions of the no-slip boundary conditionare alternatively often represented as the slip

length discussed above, the ﬁctitious distance inside the solid at which the no-slip

ﬂow boundary condition would hold; the equivalent representation of this data in

Flow rate í

peak

/D (s

-1

)

Slip length b (nm)

10 100

f *

-2

-1

Fig. 10.3. f

∗

as a function of logarithmic ﬂow rate ν

peak

/D (top panel)wheref

∗

is deﬁned

in (10.2), and the equivalent slip length (bottom panel), for deionized water (ﬁlled symbols)

and tetradecane (open symbols) between surfaces of diﬀerent levels of rms surface roughness,

speciﬁcally: 6 nm (case a; squares), 3.5 nm (case b; circles), 2 nm (case c; down triangles),

1.2 nm (case f; hexagons), 0.6 nm (case e, upper triangles) and molecularly-smooth (case

d; diamonds). The data, taken at diﬀerent amplitudes in the range of 0.3−1.5 nm and fre-

quencies in the range 6.3−250 rad/s, are mostly not distinguished in order to avoid clutter.

To illustrate the similarly successful collapse at these rough surfaces, data taken at the two

frequencies 6.3rad/s(cross ﬁlled symbols) and 31 rad/s(semi-ﬁlled symbols)forwaterare

included explicitly. After [8] with permission

524 Y. Elaine Zhu et al.

terms of the slip length is included in Fig. 10.3. While it was known previously that

a very large amount of roughness is suﬃcient to generate no-slip [22–24], an ex-

perimental study in which roughness was varied systematically [7,8] succeeded in

quantifying how much actual roughness was needed in an actual system. The criti-

cal level of 6nm considerably exceeded the size of the ﬂuid molecules. As methods

are known to achieve greater smoothness in MEMS devices, and potentially in mi-

croﬂuidic devices, this oﬀers the promise of practical applications.

Observation of rate-dependentslip suggests that ﬂuid is pinned up to some criti-

cal shear stress, beyond which it slips. However, some laboratories report slip re-

gardless of ﬂow rate [27, 29, 32–39]. Perhaps, the essential diﬀerence is that the

magnitude of shear stress is larger in the latter experiments[45]. But in cases where

slip is rate dependent, this aﬀords a potential strategy by which to eﬀect purposeful

mixing in a microﬂuidicdevice. The idea would be simply to make some patches on

the surface hydrophobic and other patches hydrophilic, so that when ﬂow was slow

enough it would be smooth, but when it was fast enough, mixing would result from

jerkiness at the hydrophobicpatches.

While it is true that slip at smooth surfaces is predicted based on computer simu-

lations [25–27,37–39], the shear rate of molecular dynamics (MD) simulations so

rms roughness (nm)

Critical shear rate g

max

-1

)

Critical stress s

max

(Pa)

62 4

( )

Fig. 10.4. Illustration that deviations from the traditional no-slip boundary condition depend

systematically on surface roughness. Here the critical shear rate for onset of slip (left or-

dinate) and critical shear stress (right ordinate) are plotted semilogarithmically against rms

surface roughness for ﬂow of deionized water (solid symbols) and tetradecane (open symbols

and semi-ﬁlled symbols as identiﬁed in [32]. The data in parentheses indicate the asymmet-

ric situation – one surface was rough and the opposed surface was atomically smooth, as

discussed in [32]. Maximum shear rate and shear stress on the crossed cylinders were calcu-

lated using known relations based on continuum hydrodynamics from [13,42]. Speciﬁcally,

max

= A

peak

and σ

peak

= 1.378ηR

1/2

peak

3/2

. After [7] with permission