Viscosity effects on hydrodynamic drainage force measurements involving deformable bodies.

Abstract

Dynamic force measurements have been made between an oil drop and a silica particle in surfactant and sucrose solutions with viscosities that range up to 50 times that of water. These conditions provide variations in the shear rate and the relative time scales of droplet deformation and hydrodynamic drainage in a soft matter system. The results obtained indicate that soft deformable boundaries have a natural response that limits the maximum shear rate that can be sustained in thin films compared to shear rates that can be attained in films bounded by rigid boundaries. In addition, to extend boundary slip studies on rigid surfaces, we use a smooth deformable droplet surface to probe the dependence of the boundary slip on fluid viscosity without the added complications of surface roughness or heterogeneity. Imposing a Navier slip model to characterize possible slip at the deformable oil-sucrose solution interface gives results that are consistent with a slip length of no larger than 10 nm over the range of solution viscosity studied, although an immobile (zero slip length) condition at the oil-sucrose solution interface is perfectly adequate. In high viscosity solutions, cantilever motion at high scan rates induces a significant cantilever deflection. A method has been developed to account for this effect in order to extract the correct dynamic force between the deformable drop and the particle.

Full text

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DOI: 10.1021/la1012473
11921
Langmuir 2010, 26(14), 11921–11927Published on Web 06/28/2010
pubs.acs.org/Langmuir
©2010 American Chemical Society
ViscosityEffectsonHydrodynamicDrainageForceMeasurementsInvolving
Deformable Bodies
Raymond R. Dagastine,*,†,‡Grant B. Webber,†,‡,3Rogerio Manica,^Geoffrey W. Stevens,†,‡
Franz Grieser,†,§and Derek Y. C. Chan†,),#
†Particulate Fluids Processing Centre,‡Department of Chemical and Biomolecular Engineering,§School of
Chemistry, and
Department ofMathematics and Statistics, University of Melbourne, Parkville, Victoria 3010,
Australia,^Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, 138632,
Singapore, and#Department of Mathematics, National University of Singapore, 117543, Singapore.
3Present address: Centre for Advanced Particle Processing, The University of Newcastle, Callaghan,
NSW 2308, Australia
)
Received March 29, 2010. Revised Manuscript Received May 26, 2010
Dynamic force measurements have been made between an oil drop and a silica particle in surfactant and sucrose
solutions with viscosities that range up to 50 times that of water. These conditions provide variations in the shear rate
and the relative time scales of droplet deformation and hydrodynamic drainage in a soft matter system. The results
obtained indicate that soft deformable boundaries have a natural response that limits the maximum shear rate that can
besustainedinthinfilmscomparedtoshearratesthatcanbeattainedinfilmsboundedbyrigidboundaries.Inaddition,
toextendboundaryslipstudiesonrigidsurfaces,weuseasmoothdeformabledropletsurfacetoprobethedependenceof
the boundary slip on fluid viscosity without the added complications of surface roughness or heterogeneity. Imposing a
Navier slip model to characterize possible slip at the deformable oil-sucrose solution interface gives results that are
consistent with a slip length of no larger than 10 nm over the range of solution viscosity studied, although an immobile
(zero slip length) condition at the oil-sucrose solution interface is perfectly adequate. In high viscosity solutions,
cantilever motionathigh scanrates inducesa significantcantilever deflection. Amethod hasbeendeveloped toaccount
for this effect in order to extract the correct dynamic force between the deformable drop and the particle.
1. Introduction
The transport of liquids of various viscosities confined near
soft surfaces is of increasing interest because of its fundamental
relevance in processes as diverse as the motility of biological
cells to the formulation and transport of structured emulsions.1
In microfluidic devices, the use of soft boundaries provides
a potential solution for drag reduction to minimize energy
expenditures2and offers a novel dimension for manipulating
localized flow.3These types of innovations are seen as crucial for
high throughput applications in which the benefits of a quanti-
tative understanding of the hydrodynamic boundary conditions
for flow at soft surfaces have not been fully explored.4The
manipulation of flow properties and boundary conditions in soft
mattersystemsoffersanewdegreeofflowcontrolthatmaynotbe
possiblewith rigid boundaries.Furthermore,changes in viscosity
affect the response time scales associated with soft material
deformations from hydrodynamic perturbations, drag reduction
schemes,2molecular and interfacial transport,5-9and hydro-
dynamic boundary conditions.2,4,6
In quantifying boundary slip using direct force measurements
with the atomic force microscope (AFM), sucrose solutions of
varying concentrations have been the liquid of choice because
they maintain Newtonian liquid behavior over a large range of
viscosities.10-15Atsolidsurfaces,flowbehaviorinregimesofhigh
shearratesinwhichboundaryslipeffectsmaybemoreprominent
canbeexploredbyincreasingthefluidviscosity. However,atsoft
boundaries,geometricdeformationsthatariseinresponsetostresses
fromtheflowfieldmayprovideanaturalregulatingmechanismthat
limittherangeofshearratesthatcanbeattained.Asaconsequence,
boundaryslipphenomenathatmaybeobservedatrigidboundaries
may not be manifested at soft interfaces.
To describe and quantify possible boundary slip at the fluid-
solid interface, the Navier slip model postulates that for a
Newtonian fluid, the tangential velocity u)at the fluid bound-
ary (with unit normal n) is proportional to the local tangential
shear stress:2,16u)= bn3[ru þ (ru)T]3(I - nn). The parameter
b, called the slip length, encapsulates details of the interaction
between the fluid and the solid boundary. At fluid-solid
boundaries, the no-slip or fully immobile boundary condition
*To whom correspondence should be addressed. E-mail: rrd@unimelb.
edu.au.
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ArticleDagastine et al.
corresponds to b = 0, while partially mobile interfaces are
characterized by a nonzero slip length.
For fluid-fluid interfaces, the effects of changing viscosity at
boundaries ofsimple soft material such as drops and bubbles can
be quantified by examining the behavior of the tangential stress
across the interface. Across an ideal clean fluid-fluid boundary
wheretheinterfaceisfullymobile,thetangentialstressisexpected
to be continuous across the interface where the velocity of the
fluidadjacenttotheinterfaceineachphaseismovingatthesame
speed. The presence of mobile surface-active materials at the
interface can provide a tangential gradient in the interfacial
tensionthatwill giverisetoajumpdiscontinuityinthetangential
shear stress.17If the gradient of the interfacial tension is suffi-
ciently high, the interface becomes immobile and the hydro-
dynamicboundaryconditionatsuchaninterfacewillbethesame
as that at a fluid-solid interface thatobeys the no-slip condition.
There is often confusion when using the terminology of a “no-
slip”boundaryconditionforafluid-fluidinterface.Forafluid-
fluid interface, a fully mobile interface, where the adjacent tan-
gentialvelocitiesofthetwofluidsareequal,satisfiesthe“no-slip”
condition,asthetangentialvelocitiesareproportionaltothelocal
tangential shear stress for Newtonian fluids. For a fluid-solid
interface, when there is a partial or full mobility at the interface,
then the tangential velocities are not proportional to the local
tangentialshearstress(sinceasolidisnotmoving,butthefluidis)
andthereisslipatasurface.Inthispaper,wedescribefluid-fluid
interfacesaseithermobileorimmobileandonlyusetheterm“no-
slip” to refer to the scenario where the tangential velocity of the
fluid-solid or fluid-fluid is zero where in both instances the
interface is immobile.
IncontrasttoallpreviousslipstudieswiththeAFM,10,11,13,15,18,19
our objective is to use the smooth deformable fluid-fluid interface
of a droplet in aqueous solution as a simple model system to
examine slip. We quantify flow conditions of fluids with a range of
viscosities when they are confined by smooth deformable bound-
aries. The known deformation characteristics of the fluid-fluid
interface and its relative sharpness allow us to make confident
estimates of the possible effects of boundary slip at smooth soft
surfaces. In particular, it offers a way to quantify boundary slip
without complications that may be due to surface roughness,10
heterogeneities,20or effects due to liquid structuring at surfaces.
Toquantifydynamiceffectsofviscosityonforcemeasurements
usingtheAFM,weinvestigatethetimedependentforcebetweena
smoothsolid silicacolloidal probe particleand asmoothdeform-
ableoildropinrelativemotioninaqueoussolutions.Thesolution
viscosity can be varied by a factor of over 50 through varying the
sucrose concentration. The quantitative behavior of the dynamic
force is determined by the boundary conditions at the solid-
sucrose solution and oil-sucrose solution interface that control
the flow of the solution in the gap between the particle and the
oil drop. However, this gap, of the order of tens of nanometers
thick,islargecomparedtothedimensionsofthesucrosemolecule
(<0.9 nm) so the aqueous solution can still be regarded as a
continuum.Weuseasilicaparticleprobeasoneoftheinteracting
surfacesbecausethehydrodynamicboundaryconditiononsucha
hydrophilic surface has now been established unambiguously.13
All experimental results reported in this paper are taken with the
same cantilever and colloidal probe particle.
2. Experimental Method
InAFMforcemeasurements,apiezoelectricactuatorisusedto
movethe end ofthecantilevertoward orawayfrom the substrate
to vary the separation X(t) (Figure 1, inset) between the particle
on the cantilever tip and the oil drop on the substrate. The
interaction between the particle and the drop will cause the
deflection of the cantilever to change as the cantilever is moved.
This deflection can be converted to the force between the particle
andthedropviaHooke’slaw,oncethecantileverspringconstant
has been determined. However, in regimes of high scan rates and
at high viscosities, the cantilever will also deflect as a result of the
hydrodynamic drag exerted on it as it is being driven by the
piezoelectricactuator.Wehavedevelopedamodeltocharacterize
cantilever deflections that result from hydrodynamic drag that is
separate from deflections due to interaction between the particle
andthedrop.Thisisparticularlyimportantsincethepiezoelectric
actuator actually moves the cantilever with a variable velocity
with the result that such deflections can potentially be misinter-
preted as a variable force between the particle and the drop.
Anatomicforcemicroscope(AsylumResearch,SantaBarbara,
CA)wasusedtomeasurethedynamicforcebetweenatetradecane
oil drop in aqueous sucrose solution with a silica colloid probe
particleofradiusRp=25(2μmgluedontheendofacantilever
(Veeco MLCT Series) whose spring constant, K = 0.039 ( 0.004
N/m, was calibrated by the Hutter-Bechhoefer method.21The
shear viscosity, μ, of the sucrose solution was varied by up to
50 times that of water by increasing the sucrose concentration to
60%. The sucrose solution also contained 5 mM of the anionic
surfactant sodium dodecyl sulfate (SDS), which is below the
criticalmicelleconcentrationof8mM.Forcemeasurementswere
takenwiththeprobeparticlecarefullyalignedwiththeapexofthe
sessile oil drop of undeformed radius Rd= 107 ( 2 μm on the
substrate.
Aforcemeasurementruncomprisedanapproachbranch,asthe
cantilever was moved toward the oil drop by piezoelectric actua-
tor, followed by a retract branch, when the cantilever was moved
away from the drop. The position, X(t) (Figure 1, inset), of the
cantilever was monitored by using a linear variable differential
transformer (LVDT). The nominal speed or scan rate of the
piezoelectric actuator was varied from less than 1 μm/s up to
20 μm/s. At a set scan rate, the actuator velocity was observed to
Figure 1. Measured forces (∼1% of acquired points are shown)
between a silica colloid probe and the tetradecane oil drop in
sucrose solution with 5 mM SDS versus relative cantilever dis-
placement,ΔXatascanrateof1μm/s.Experimentsarecompared
to predictions of the Stokes-Reynolds-Young-Laplace model
with the immobile boundary condition at the oil drop (---)
andtheanalytichighforceformulagivenbyeq8(---).Forclarity,
results at different sucrose concentrations have been offset verti-
cally. The upper (lower) data points of each set are the forces on
approach(retract).Inset:Schematicdiagramofthecantileverwith
colloid probe and oil drop on the substrate.
(17) Levich, V. Physicochemical Hydrodynamics; Prentice Hall: London, 1962.
(18) Vinogradova, O. I. Langmuir 1996, 12, 5963–5968.
(19) Vinogradova, O. I.; Yakubov, G. E. Phys. Rev. E 2006, 73, 045302/1–
045302/4.
(20) Chan,D.Y.C.;Uddin,M.H.;Cho,K.L.;Liaw,I.I.;Lamb,R.N.;Stevens,
G. W.; Grieser, F.; Dagastine, R. R. Faraday Discuss. 2009, 143, 151–168.
(21) Hutter, J. L.; Bechhoefer, J. Rev. Sci. Instrum. 1993, 64, 1868–1873.

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Langmuir 2010, 26(14), 11921–11927
Dagastine et al. Article
vary by up to (50% during the approach and retract runs (see
Figure 2), so values of the instantaneous velocity, dX(t)/dt, were
used in all data analysis and modeling.
Timevariationsofthecantileverdeflectioncanbeconvertedto
the interaction force between the colloidal particle probe and the
tetradecanedrop.However,athighscanratesandhighviscosities,
the cantilever also deflects as a result of its motion. Such deflec-
tions are additional to that caused by interaction between the
particle probe and the drop, and they need to be subtracted from
the measured total cantilever deflection in order to derive the
dynamic force between the particle and the drop.
TheSDSadsorbsontothesurfaceofthetetradecanedropsand
creates a negative surface charge.22Thus, there was a repulsive
electric double layer repulsion between the oil drop and the
negativesilicaparticlewhichwassufficienttopreventparticle-oil
drop coalescence. The range of the repulsive electrical double
layer interaction was not expected to affect the long-ranged
hydrodynamicinteraction.23TheadsorbedSDSwasalsoexpected
to dominiate the influence of low concentrations of adventitious
surface active species that generally exist at oil-water interfaces.
The surface potential of the silica sphere as a function of sucrose
concentration was determined by fitting a electrical double layer
model to the sphere-plate AFM force measurements performed
using silica spheres from thesame batch as the spheres used in the
droplet measurements. The tetradecane droplet zeta potential is
based on electrokinetic measurements in SDS solutions.22These
valuesarenotexpectedtodeviatesignificantly,astheSDSismuch
more surface active than the sucrose.
Properties of the system and parameters used in analyzing
experimental results are given in Table 1. The pendent drop
technique was used to measure interfacial tensions with SDS up
to 30% sucrose, but the method was not accurate at high
viscosities. Therefore, the interfacial tension of the 5 mM SDS
and40%or60%sucrosewasestimatedusingthehighforceresult
in eq 8 (see later). These estimates are also consistent with an
extrapolation of a Frumkin isotherm model fit to the lower
concentration sucrose-SDS data.
3. Theoretical Analysis
As the particle and drop dimensions are many tens of micro-
meters whereas interaction between the particle and the oil drop is
onlysignificantwhentheirseparationisofordertensofnanometers,
the hydrodynamic problem can be treated in the lubrication ap-
proximation.25Theexperimentalsetuphasaxialsymmetryandcan
be described by coordinates (r, z) so the time evolution of the
separationh(r,t) betweentheparticleandtheoildrop(seeFigure1,
inset) can be described by the Stokes-Reynolds equation:26
∂h
∂t¼
1
12μr
∂
∂r
rh4þðbdþbpÞh3þ12bdbph2
bdþbpþh
"#
∂p
∂r
0
@
1
A
ð1Þ
where p(r,t) is the hydrodynamic pressure in the sucrose solution
betweentheoil dropandtheparticle.Whereasthis equationallows
for different Navier slip lengths at the particle, bp, and at the oil
drop,bd,wewillsetbp=0sinceweknowthattheno-slipcondition
holds at the particle surface.13
The separation between the particle and the oil drop during
thecourseofthedynamicinteractionisdescribedbytheYoung-
Laplace equation:27
σ
r
∂
∂r
r∂h
∂r
??
¼ 2σ
1
Rdþ1
Rp
!
-ðpþΠÞð2Þ
where the oil-sucrose solution interfacial tension, σ, is assumed
to be constant and the disjoining pressure, Π(r,t), accounts for
surfaceforcessuchaselectricaldoublelayerrepulsionandvander
Waals attraction that are of relatively short-range compared to
hydrodynamic interactions. Equations 1 and 2 are both referred
toastheStokes-Reynolds-Young-Laplace(SRYL)equations,
which together with appropriate initial and boundary conditions
can be solved bythe method of lines25,28from which the dynamic
force,F(t),betweentheparticleandtheoildropcanbecalculated:
FðtÞ ¼ 2π
Z¥
0
r½pðr,tÞþΠðr,tÞ?dr
ð3Þ
Axialsymmetryconsiderationsrequirethespatialderivativeof
the pressure and the separation to be zero at r = 0: ∂p/∂r = 0 =
∂h/∂r. In the lubrication limit, the initial separation between the
particle and the undeformed drop has the parabolic form,
hðr,t ¼ 0Þ ¼ hoþr2=Ro, Ro-1? Rp-1þRd-1
??
ð4Þ
Figure 2. Leftaxis:Timedependenceofthemeasured(b,∼1%of
theacquireddataareshown)andpredicted(reddashedline)appa-
rent force, Fapparent, and the predicted interaction force, F, accord-
ing to the SRYL model with immobile boundary conditions (blue
solidline)andwithasliplengthof10nmattheoil-sucrosesolution
interface(bluedashedline) atascan rateof20μm/sin40%sucrose
solutionwith5mMSDS.Rightaxis:PiezoelectricdrivespeeddX/dt
(pinkdashedline)andcentralfilmvelocitydh(0,t)/dt(tealsolidline).
Table1.PropertiesoftheExperimentalSystemandParametersUsed
in Modeling at 5 mM SDS and Different Sucrose Concentrations
sucrose (%)
interfacial tension, σ ((2 mN/m)
viscosity, μ (Pa s)24
silica sphere surface potential
((10 mV)
tetradecane droplet zeta potential22
((10 mV)
spring constant, K ((0.004 N/m)
radius of drop, Rd((2 μm)
radius of sphere, Rp((2 μm)
contact angle of drop,23θo((5?)
aDerived by fitting using eq 8.
0
16
0.001
-40
20
8.0
0.0019
-25
40
9.0a
0.0058
-25
60
10a
0.052
-25
-100100
-100
-100
0.039
107
25
50?
(22) Nespolo, S. A.; Bevan, M. A.; Chan, D. Y. C.; Grieser, F.; Stevens, G. W.
Langmuir 2001, 17, 7210–7218.
(23) Webber, G. B.; Manica, R.; Edwards, S. A.; Carnie, S. L.; Stevens, G. W.;
Grieser,F.; Dagastine,R.R.;Chan, D.Y.C.J.Phys.Chem.C2008,112,567–574.
(24) Green, D. W.; Perry, R. H. Perry’s Chemical Engineering Handbook;
McGraw-Hill: New York, 2008.
(25) Carnie, S. L.; Chan, D. Y. C.; Lewis, C.; Manica, R.; Dagastine, R. R.
Langmuir 2005, 21, 2912–2922.
(26) Manica, R.; Connor, J. N.; Carnie, S. L.; Horn, R. G.; Chan, D. Y. C.
Langmuir 2007, 23, 626–637.
(27) Manica, R.; Connor, J. N.; Dagastine, R. R.; Carnie, S. L.; Horn, R. G.;
Chan, D. Y. C. Phys. Fluids 2008, 20, 032101/1–032101/12.
(28) Dagastine,R.R.;Manica,R.;Carnie,S.L.;Chan,D.Y.C.;Stevens,G.W.;
Grieser, F. Science 2006, 313, 210–213.

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Article Dagastine et al.
The initial distance of closest approach, ho, between the particle
and the drop can be set experimentally to be in the range of
1-6μminaforcerun.Itsprecisevaluecanbedeterminedtowithin
(0.01 μm by requiring the model to fit the entire force curve. Far
fromthecentralaxis,whererf¥,thepressurevanishesasp∼r-4
and this condition is implemented as27r(∂p/∂r) þ 4p = 0.
For measurements in low viscosity liquids, the dynamic force,
F, between the particle and the drop is obtained from the mea-
sured cantilever deflection, S, and the cantilever spring constant,
K,byapplyingHooke’slaw:F=KS.However,athighscanrates
and in high viscosity solutions, the cantilever also deflects due to
hydrodynamic drag when it is being moved by the piezoelectric
actuator.Weexpectthat,underStokesflow,thisdeflection,Sdrag,
will be proportional to the actuator velocity, dX/dt, and the
viscosity, μ, of the solution and inversely proportional to the
cantilever spring constant, K:
Sdrag ¼ -Cμ
K
dX
dt
ð5Þ
The sign convention in eq 5 is such that a positive deflection
correspondstoanapparentrepulsionbetweentheparticleandthe
drop. The constant, C, can be determined from the force curve at
largeseparationswhendropdeformationisnegligibleanditsvalue
will depend only on the geometric properties of the cantilever.
Following earlier work,25,28-33we use the constant volume
constraint to obtain the following boundary condition at the
outer limit of the solution domain, r = rmax, if the three phase
contact line of drop on the substrate is assumed to be immobile:
∂h
∂t¼dX
?
dt-Cμ
K
d2X
dt2þ1
!
K
dF
dt
??
-
1
2πσ
dF
dt
?
1þ1
2log
rmax2
4Rd2

þ1
2log
1þcos θo
1-cos θo
??
2
4
3
5ð6Þ
Here, θois the contact angle of the undeformed drop on the sub-
strate.Ontheotherhand,ifthedroponthesubstratemaintainsa
constant contact angle, θ, this boundary condition would be
∂h
∂t¼dX
dt-Cμ

K
d2X
dt2þ1
K
dF
dt
??
-
1
2πσ
dF
dt
??
1þ1
2log
rmax2
4Rd2
!
þ1
2log
1þcos θ
1-cos θ
??
-
1
2þcos θ
2
4
3
5
ð7Þ
The derivations of eqs 6 and 7 are given in the Appendix.
However, we should note that cantilever drag also affects the
form of the large r boundary condition, eq 6 or 7, for the SRYL
equations that determine the dynamic interaction between the
particle and the deformable drop.
At low scan rates and in the regime when the repulsive force
betweentheparticleandthedropishighsothattheapproachand
retract force data overlap, the force, F, versus relative cantilever
displacement, ΔX (up to an additive constant), has the following
approximate analytic form27
ΔX ∼
F
4πσ
log
FRo
8πσRd2
??
þlog
1þcos θo
1-cos θo
??
þ1
"#
ð8Þ
This result is independent of viscosity or details of the disjoining
pressure as long as it is sufficiently repulsive to prevent coales-
cence. It provides a useful check between theory and experiment.
Furthermore, the interfacial tension, σ, of the drop determined
from this result can be compared to values that have been
measured independently.23,34
4. Results and Discussion
Low Scan Rates, Variable Viscosity. Results for the force,
F, versus relative cantilever position, ΔX, at a low scan rate
(1 μm/s) are shown in Figure 1 for 0%, 20%, and 40% aqueous
sucrose solutions, where, for clarity, only about 1% of measured
data points have been plotted. At this low scan rate, cantilever
deflection due to hydrodynamic drag is negligible. However, the
almost 6-fold increase in solution viscosity over this sucrose
concentration range enhances the hydrodynamic interaction
and gives rise to the growth of the size of the hysteresis loop
between the approach and retract branches of the force. At these
low scan rates, the SRYL model gives excellent agreement bet-
ween theory and experiments, particularly with regards to the
regionaroundtheforce hysteresis usingthe no-slipand immobile
hydrodynamic boundary condition, corresponding zero slip
lengths: bd= 0 = bp, at the particle and at the deformable
tetradecane drop. Earlier AFM studies9,23,28,34,35examining the
presenceofMarangoniflows,electrokineticmeasurements,22,35,36
orlargerscaledrainagestudiesstudies37,38haveshownthatavery
low concentration of surfactant is sufficient to give rise to an
immobile interface. In this work, at the concentration of 5 mM
SDS in the sucrose solution, which is below the CMC value of
8 mM, the surface coverage is around 90% saturation22and thus
an immobile interface is expected for this oil and surfactant
system.
In the high force region (>2 nN) where the approach and
retract data overlap, the force, F, as a function of displacement,
ΔX, dependence can be fitted to the high force analytic result
(dashed lines) given in eq 8.26,27The fitting procedure also allows
the interfacial tension of the tetradecane drop at various sucrose
concentrations to be determined (Table 1). The interfacial ten-
sions obtained this way agreed with available values determined
by the pendent drop method. This high force analytic result does
not depend on the detailed form of the repulsive disjoining
pressure,Π(r,t),aslongastheparticleandthedropareprevented
from coalescing.
High Scan Rates, Variable Viscosity. At higher scan rates,
and particularly at high solution viscosities, it is important to
quantify separate contributions to the observed deflection, S, of
thecantileverduetointeractionbetweentheparticleandthedrop
and due to hydrodynamic drag on the cantilever. Significant
cantilever deflection due to hydrodynamic drag is evident even
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4336–4344.

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Dagastine et al.Article
whenthecolloidprobeisfarfromtheoildrop,particularlyathigh
viscosities. Hitherto, this drag force has been assumed to be
constantanditsvalueisestimatedbyusinganadditiveconstantto
enforceazeroforceatlargeseparations.13,39However,linearityof
thehydrodynamicproblemsuggeststhatthecantileverdragforce
should be proportional to the velocity of the cantilever and the
solution viscosity. In section 3, we indicate that cantilever drag
has to be taken into account in formulating an appropriate
boundary condition for the SRYL model.
We can define the measured apparent force, Fapparent, as the
product of the observed cantilever deflection, S, and the cantilever
spring constant, K. This can be written as the sum of a separation-
independentcantileverdragforce,Fdrag,andtheinteractionforce,F:
Fapparent? KS ¼ FdragþF ¼ -KSdragþF
¼ Cμ dXðtÞ=dtþF
WedeterminethevalueoftheconstantC=(6.0(0.6)?10-3mfor
our cantilever, from experimental results in a 40% sucrose solution
at 20 μm/s during 0 < t e 0.04 s (Figure 2) when the particle is far
from the oil drop and deformation is negligible. In this regime, the
analyticalformofF=-(6πμRo2/h)(dX/dt)(withoutdeformations)
can be used. The value of C so obtained is then used to analyze all
other data obtained with the same cantilever at different scan rates
and different viscosities.
In Figure 2, we compare experimental apparent force data,
Fapparent, in a 40% sucrose (with 5 mM SDS) solution at 20 μm/s
predictedusingthesolutionoftheSRYLmodelgivenineqs1-6.
We see excellent quantitative agreement with the no-slip bound-
ary condition at the oil-water interface. From this comparison,
we can extract the interaction force, F, between the particle and
the deformable drop. The effects of the cantilever drag are
significant in regimes where the film drainage velocity is high,
as measured by the rate of change of the central film thickness,
dh(0,t)/dt.FromFigure2,thefilmdrainageratesarehighnearthe
rise of the repulsive region during the approach phase and near
the hydrodynamic attractive minimum during the retract phase.
TheeffectofusingaNavierslipmodelattheoil-sucrosesolution
interface with a slip length of10 nm onthe interaction force, F, is
also shown in Figure 2. Although the level of precision of the
experimental results can accommodate a slip length between 0
and 10 nm, we have no other compelling reason to conclude that
slip occurs in our system.
In Figure 3, we compare experimental apparent force data,
Fapparent,andpredictionsfromtheSRYLmodelatthescanrateof
20 μm/s for three different sucrose concentrations that span a
6-fold change in the viscosity. Similar comparisons at lower scan
ratesshowequallygoodagreementbetweenexperimentandtheory,
again with the immobile boundary condition on the oil drop.
The shear rateattheoil-sucrose solution interfaceatdifferent
radialpositionsinthefilmofsucrosesolutionbetweentheparticle
andtheoildropisshownintheinsetofFigure3correspondingto
different times marked on the force curve during particle-drop
interaction in a 40% sucrose solution. The sign of the shear rate
depends on whether the sucrose solution is being expelled from
between the oil drop and particle during the approach phase or
being sucked in to fill the space between the oil drop and the
particle during the retraction phase. The magnitude of the shear
rate remains at less than 6200 s-1. The reason for the modest
magnitude of shear rate is that the deformability of the oil drop
provides a natural limiting mechanism. As the particle is driven
toward the drop, the repulsive hydrodynamic pressure will
ð9Þ
increase. When this pressure approaches the Laplace pressure
of the drop, the oil-sucrose solution interface will begin to
deform and flatten, and thereby increase the effective interaction
area between the particle and the flattened drop. This enhanced
hydrodynamic repulsion is sufficient to prevent the particle from
further approaching the oil-water interface. We can see this from
the calculated rate of change of the central separation at r = 0:
dh(0,t)/dt, between the drop and the particle shown in Figure 2
(continuous line, right ordinate). During the period 0.10 s < t <
0.16s,dh(0,t)/dtissmallsotheseparationbetweentheparticleand
the drop is nearly constant while the force changes significantly in
both magnitude and sign. These changes are accompanied by
changes in the flow direction of the intervening sucrose solution
as reflected in the change in sign of the shear rate (Figure 3, inset).
Interestingly, after the force reaches the attractive minimum
duringtheretractionphaseattg0.16sinFigure2,thesurfaceof
the oil dropseparates very quickly from the particle and attains a
local speed of ∼60 μm/s, almost twice that of the speed of the
piezoelectric actuator. At the corresponding force minimum in
Figure 3, point C, the shear rate has the largest magnitude
(Figure 3, inset).
Highest Solution Viscosity. At the highest sucrose concen-
tration of 60% (Figure 4) where the solution viscosity is over 50
timesthatofwater,wecanonlydrivethecantileveratamaximum
speed of 3 μm/s as we need to keep the cantilever response within
the linear regime. To demonstrate the effects of incorporating a
nonzero slip length at the oil-sucrose solution interface, we also
show calculated values of the interaction force that include slip
lengths of 10, 100, and 1000 nm at the oil-sucrose solution
interface in Figure 4. From this, we conclude that if we wish to
allowforthepossibilityofanonzerosliplengthatthedeformable
tetradecane-sucrose solution interface, a slip length of 10 nm
wouldbeanupperbound.Indeed,thepresentcomparisonbetween
theory and experiment does not provide compelling evidence of a
nonzero slip length. Similar comparisons at other sucrose concen-
trations lead to the same conclusion (see also Figure 3).
Additional Observations. From the results in Figures 2-4,
we note the following:
(i) For times t g 0.7 s, which is just after the force
maximum at point A in Figure 4, the retraction
phase has commenced and the particle is being
pulled away from the oil drop by the piezoelectric
actuator.Nevertheless,weseeintheinsetofFigure4
that because the oil drop is deformable, the sucrose
Figure 3. Time dependence of the experimental apparent force,
Fapparent (points), at three different sucrose concentrations (in
5 mM SDS) and at a scan rate of 20 μm/s. Comparison with the
SRYLmodelwithimmobileboundaryconditionsattheoil-sucrose
solution interface (lines). Inset: Shear rate at the oil-sucrose solu-
tioninterfaceat40%sucroseasafunctionoftheradialcoordinate,r,
corresponding to times keyed to the force curve in the main figure.
(39) Craig, V. S. J.; Neto, C. Langmuir 2001, 17, 6018–6022.

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DOI: 10.1021/la1012473Langmuir 2010, 26(14), 11921–11927
Article Dagastine et al.
solution film between the particle and the oil drop
continues to thin until point C (t ∼ 0.9 s) where the
sucrosefilmreachesaminimumthicknessofaround
h ∼ 25 nm (curve C inset Figure 4).
As the sucrose solution is over 70 nm thick around
point D (Figure 4), the physical origin of the force
minimumisthereforedueentirelytohydrodynamic
interactions across a film of sucrose solution con-
fined between the silica particle and the deformed
surface of the oil drop. Repulsive electrical double
layerinteractionsandattractivevanderWaalsforces
are negligible at such separations. For such thick
films (compared to the size of the sucrose molecule,
∼0.9 nm), the sucrose solution can certainly be
treated as a continuum Newtonian fluid character-
ized by a constant shear viscosity.
In allowing for a slip length of up to 10 nm at the
deformable oil-sucrose solution interface, the dis-
cernible difference with the immobile interface result
occurs at around the force minimum, point C in
Figure3andpointDinFigure4.Theshearrateinthe
film at these points is large (Figure 3, inset) because
the deformable oil-sucrose solution interface is re-
cedingrapidlyfromtheparticle(Figure2,fordh(0,t)/
dt, continuous curve, right ordinate). Thus, with the
addeddegreeoffreedomofdeformability,weseethat
theeffectsofboundaryslipismostnoticeableathigh
shear rates, which does not necessarily occur when
the film thickness is at a minimum as expected for
flowbetweenrigidboundaries.Thehighestshearrate
therefore occurs at a point controlled by a combina-
tion surface deformability, effective area of interac-
tion, surface separation, and interfacial velocity.
The detailed analyses developed here for the hydro-
dynamic drag on the cantilever have up to a 15%
effect on measurements at higher viscosities and
velocities.Itisinterestingtonotethatmostslipstudies
using an AFM on rigid surfaces have not considered
accounting for the deflection of the cantilever in this
fashion, even though recent studies that examine slip
between hydrophilic surfaces report anomalous dif-
ferences between different types of cantilevers12with
no definitive explanations, but where the character-
istic drag on the cantilever may differ.
(ii)
(iii)
(iv)
(v) TheuseofaNavierslipmodelatthesurfaceoftheoil
drop involves a further simplification. If there are
mobile surfactants at the oil-sucrose solution inter-
face, one should take into account hydrodynamic
flow both in the sucrose solution in between the oil
drop and the particle as well as in the interior of the
oil drop. These two flow fields will be coupled at the
interfacebyatangentialstressbalanceconditionthat
involves local tangential gradients of the interfacial
tension and convection and diffusion of the mobile
surfactants.6,9Whereas the formulation of such a
detailed model is available, the additional technical
complexity involved in implementing this model is
quitesignificant.7Thepresentstudyusingthesimpler
Navier slip model does not provide compelling evi-
denceforasliplengthlarger than10nm,orabout 10
timesthedimensionsofthesucrosemolecule.Indeed,
a zero slip length is still able to describe the experi-
mental results very well. In addition, mobility of the
surfaceisunlikelygivenwehaveapproximately90%
saturation of the oil-sucrose solution interface with
surfactant. In a noncreeping flow regime, where
highershearratesmaybepossible,interfacemobility
may become a more important factor.
5. Conclusions
Using a well-characterized system of an oil drop in a sucrose
solution as a model soft smooth interface for which the dynamic
behaviorisdominatedbyhydrodynamicinteractions,weareable
to demonstrate that boundary slip can manifest as a Navier slip
lengthofnomorethan10nmorabout10timesthemolecularsize
of the sucrose molecule under the combined effects of interfacial
deformationsanda50-foldvariationinsolventviscosity.Thisisa
reasonable expectation since the slip length should reflect char-
acteristic length scales at the fluid interface. The ability of the oil
drop to deform suggests that soft systems provide a natural
mechanism to limit shear, which will have implications in the
possibilityofusingsoftmaterialtominimizeenergyrequirements
in massively parallel microfluidic operations.
Acknowledgment. This work is supported in part by the
Australian Research Council (ARC) Discovery Grant Scheme
and together with AMIRA International and the State Govern-
mentsofVictoriaandSouthAustraliathroughtheARCLinkage
grant scheme. D.Y.C.C. is an Adjunct Professor at the National
University of Singapore.
Appendix
The total deflection, S, of the cantilever is caused by fluid drag
arising from cantilever motion (dX/dt) and by the interaction
Figure 4. Time dependence of the experimental apparent force,
Fapparent(b), the interaction force, F, from the SRYL model with
no-slip boundary conditions (blue solid line) and with slip lengths
ranging between 10 to 1000 nm at the oil-sucrose solution inter-
face (blue dashed line). Inset: Separation between the deformable
oil-sucrose solution interface and the particle as a function of the
radial coordinate, r, corresponding to times keyed to the force
curve in the main figure.
Figure 5. Schematic diagram of the particle-drop geometry in the
atomic force microscope.

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Langmuir 2010, 26(14), 11921–11927
Dagastine et al. Article
force,F,betweenaparticleanddrop.Thefluiddragcontribution
to the deflection, Sdrag, will be proportional to the piezoelectric
actuator velocity, dX/dt, and the viscosity, μ, of the solution and
inversely proportional to the cantilever spring constant, K:
S ¼ SdragþF=K ¼ -Cðμ=KÞðdX=dtÞþF=K
whereS>0correspondstorepulsionandtheconstantCdepends
on the cantilever geometry.
From the geometry of schematic atomic force microscope in
Figure 5, we have
ðA1Þ
X þS ¼ zpþhþzd
ðA2Þ
Differentiationwithrespecttotandnotingthatparticleshape,zp,
is constant gives
dh=dt¼ dX=dtþdS=dt-0-dzd=dt
¼ dX=dt-Cðμ=KÞðd2X=dt2Þþð1=KÞðdF=dtÞ-dzd=dt
ðA3Þ
If the three phase contact line of the drop on the substrate is
pinned at a constant position during interaction, then at a
position r = rmaxthat is outside the interaction zone the drop
shape has the form given by eq 21 of Carnie et al.25
?
dzd
dt
¼
1
2πσ
dF
dt
?
1þ1
2log
rmax2
4Rd2
!
þ1
2log
1þcos θo
1-cos θo
??
ðA4Þ
2
4
3
5
where θois the contact angle of the undeformed drop on the
substrate. Thus, the boundary condition at r = rmax, with canti-
lever drag with the pinned contact line boundary condition, is
∂h
∂t¼dX
?
dt-Cμ
K

d2X
dt2þ1
!
K
dF
dt
??
?
-
1
2πσ
dF
dt
?
1þ1
2log
rmax2
4Rd2
þ1
2log
1þcos θo
1-cos θo
?
2
4
3
5
ðA5Þ
On the other hand, if the contact angle the drop on the
substrate is constant during interaction, the boundary condition
at r = rmaxcan be deduced from eq 25 of Carnie et al.25to be
dh
dt¼dX
dt-Cμ

K
d2X
dt2þ1
K
dF
dt
??
-
1
2πσ
dF
dt
??
1þ1
2log
rmax2
4R2
!
þ1
2log
1þcos θ
1-cos θ
??
-
1
2þcos θ
2
4
3
5
ðA6Þ
The relative magnitude of the new term involving (d2X/dt2)
term in eqs A5 and A6 can be estimated by the standard scaling
usedinthisproblemintermsofthecapillarynumber:Ca=μV/σ,
where V is the scan rate
h ∼ RoCa1=2; t ∼ ðRoCa1=2Þ=V;
r ∼ RoCa1=4; F ∼ ðσ=RoÞðRoCa1=4Þ2
With this scaling, the various terms in eq A5 have the order:
dh
dt∼dX
dt∼
1
2πσ
dF
dt
??
1þ1
2logrmax2
4Rd2
!
þ1
2log
1þcos θo
1-cos θo
??
2
4
3
5∼Oð1Þ
1
K
dF
dt
??
∼ Oðσ=KÞ
Cμ
K
d2X
dt2∼ OðC=RÞðσ=KÞCa1=2
Using the estimated value of C ∼ 6 ? 10-3m from experimental
dataandusingupperboundestimatesforμ∼5?10-3PasandV
∼ 20 ? 10-6m/s and system parameters R ∼ 50 ? 10-6m and
system parameters σ ∼ 50 ? 10-3N/m, K ∼ 40 ? 10-3N/n, we
find
Cμ
K
d2X
dt2∼ OðC=RÞðσ=KÞCa1=2∼ 0:14
which is around a 14% effect, assuming the scaled (d2X/dt2) is of
order 1. The effect of this cantilever drag term will be smaller at
lower viscosities or scan rates.
In implementing this cantilever drag contribution to the
boundary condition at rmax, we use the cantilever displacement
function, X(t), measured by the LVDT in the AFM. This
function, available as a list of data points, exhibits a sharp jump
inthevalueofdX(t)/dtatthecommencementoftheretractphase.
As a result, the value of d2X(t)/dt2has an apparent jump
discontinuity which is simply an artifact of the discrete nature
of data trace of X(t). This then results in a small jump disconti-
nuity in the predicted value of the apparent force just after the
commencement of the retraction phase. This technical deficiency
can be corrected if finer sampling of the displacement function,
X(t), is available.