Slow, non-diffusive dynamics in concentrated nanoemulsions

0.22 MB·English

by  H. Guo

Slow, Non-Diffusive Dynamics in Concentrated Nanoemulsions
H. Guo,1 J. N. Wilking,2 D. Liang,1 T. G. Mason,2,3 J. L. Harden,4 and R. L. Leheny1
1Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218
2Department of Chemistry and Biochemistry, University of California-Los Angeles, Los Angeles, California 90095
3Department of Physics and Astronomy, University of California-Los Angeles, Los Angeles, California 90095
4Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
(Dated: February 6, 2008)
7
0 Usingmultispecklex-rayphotoncorrelationspectroscopy,wehavemeasuredtheslow,wave-vector
0 dependentdynamicsofconcentrated,disordered nanoemulsions composed ofsilicone oil dropletsin
2 water. The intermediate scattering function possesses a compressed exponential lineshape and a

n relaxation time that varies inversely with wave vector. We interpret these dynamics as strain in
a responsetolocalstressrelaxation. Themotionincludesatransientcomponentwhosecharacteristic
J velocity decays exponentially with time following a mechanical perturbation of the nanoemulsions
3 andasecondcomponentwhosecharacteristicvelocityisessentiallyindependentoftime. Thesteady-
2 statecharacteristicvelocityissurprisinglyinsensitivetodropletvolumefractionintheconcentrated
regime, indicating that the strain motion is only weakly dependent on the droplet-droplet interac-

] tions.
t
f
o PACSnumbers: 82.70.Kj,62.25.+g,64.70.Pf,61.10.Eq
s
.
t
a I. INTRODUCTION makethetechniqueideallysuitedforprobingthemotion
m of the nanoscale droplets in the concentrated, jammed
- regime. We find these dynamics are described by ex-
d An emulsion is a dispersion of a liquid within an-
tremely slow ballistic motion that can be modeled in
n other, immiscible liquid. In most cases, emulsions are
terms of strain in response to localized changes in in-
o not thermodynamically stable but rather are long-lived,
ternal stress. Such dynamics have been observed in a
c metastable states formed through arrested phase sepa-
[ wide range of disordered soft-solid materials including
ration or mechanical mixing [1]. In the formation of
dilute and concentratedcolloidal gels [5, 6], clay suspen-
1 emulsionsthroughmixing,dropsofoneliquidarebroken
sions [7, 8], dense ferrofluids [9], concentrated emulsions
v apart within another liquid through strong shear flow.
of micrometer-scale droplets [10], and a polymer-based
0 For emulsions comprised of low viscosity liquids, such as
4 sponge phase [11]. In contrast to most previously stud-
oil in water, the shear flows accessible in common mix-
5 ied systems, in which these dynamics typically evolve in
ing devices can produce micrometer-scale droplets that
1 a manner akin to aging in glasses, the slow dynamics
are stabilized against coalescence by added surfactant.
0 in the nanoemulsions display a steady-state component.
7 However,withextremelystrongshearonecancreatena-
The presence of steady-state strain motion in the na-
0 noemulsions containing droplets with diameters well be-
noemulsionsthussuggestsaprocedureforunderstanding
/ low100nm,asrecentworkemployinghigh-pressureflows
t better these dynamics and their significance by investi-
a hasdemonstrated[2]. Atlowconcentrations,thedroplets
m gatingsystematicallytheirrelationshiptoothermaterial
in nanoemulsions behave like spherical colloids that dif-
properties. As an initial effort, we have characterized
- fuse freely. At higher concentrations, the droplets enter
d the dependence of the dynamics on droplet volume frac-
a jammed state, and the nanoemulsions behave as dis-
n tion. Surprisingly,wefindthestraindynamicsarelargely
ordered elastic solids. However, due to the deformable
o insensitive to changing volume fraction within the con-
c nature of the droplets, the structure of concentrated na-
centratedregimedespitethelargechangeinmacroscopic
: noemulsionscandeviatesignificantlyfromthatoftypical
v elastic modulus that accompanies the change.
colloidal glasses [3, 4]. For example, the volume frac-
i
X tion of droplets in concentrated nanoemulsions can ex-
r ceedthe randomclosepackedlimit for spheres,implying
a
faceteddropletshapeslikeinamacroscopicfoam. Thus, II. EXPERIMENTAL PROCEDURES
in many respects concentrated nanoemulsions represent
a new type of nanostructured soft material. Thenanoemulsionswerepreparedfollowingprocedures
While the structural properties of concentrated na- described elsewhere [2] and consisted of poly(dimethyl-
noemulsionshaverecentlybeenelucidatedthroughsmall siloxane) oil droplets in water stabilized by sodium do-
angle scattering studies [3, 4], less is understood about decyl sulfate (SDS). Two sets of samples were included
the nature of the droplet dynamics in this glassy state. in the study, one with an average droplet radius a =
To investigate these dynamics, we have performed x- 46 nm, which we label set A, and one with a = 36
rayphotoncorrelationspectroscopy(XPCS)experiments nm, which we label set B. The volume fraction φ of oil
on a series of concentrated nanoemulsions. The large droplets was varied over the ranges 0.33 < φ < 0.55
wave vectors and long time scales accessible with XPCS for A and 0.28 < φ < 0.45 for B. The upper limits
2
of these ranges were set by the volume fractions of the severalhoursforeachsampleinordertocharacterizeany
stock emulsions produced in the synthesis. To obtain evolution of the structure or dynamics of the nanoemul-
lower φ, we diluted the stock emulsions with a surfac- sions. As an additional part of the protocol to avoid
tant solution identical to the continuous phase: 10 mM radiation damage to the samples, each measurement of
SDS in water. Due to the ionic nature of SDS, the sur- g (q,t) was performed with the x-ray beam incident on
2
faces of the oil droplets are negatively charged, and the a new region of the sample.
droplets interact through a Coulombic repulsion with a
Debye screeninglength that is anappreciable fractionof
the droplet radii. To account for the range of interac- III. RESULTS AND DISCUSSION
tion between droplets, an effective droplet volume frac-
tion can be estimated as φ ≈φ(1+h/a)3 [12], where A. Scattering Intensity and Dynamic Structure
eff
hisaneffectivedroplet“shell”thicknesswhosesizeises- Factor
timated from the average inter-droplet separation using
an interactionpotential for repulsive droplets [1, 13, 14]. Figures 1(a) and 1(b) show the x-ray scattering inten-
This renormalization of volume fraction leads to fluid- sityI(q)forananoemulsionfromsetAwithφ =0.82
eff
solid transitions in nanoemulsions at φeff ≈ 0.58, near and one from set B with φeff =0.60, respectively, mea-
the colloidal glass transition, independent of droplet ra- sured at several waiting times t since the nanoemul-
w
diusandfurtherleadstoascalingoftheshearmodulusof sions were loaded into sample cells. The measured I(q)
concentratednanoemulsionsofdifferentradiiontoamas- are unnormalized; therefore, to compare measurements
ter curve as a function of φeff [14]. The corresponding performed on different regions of the samples, we arbi-
rangesofeffectivevolumefractionsforthenanoemulsions trarily normalize each set of curves with respect to the
in the XPCS measurements were 0.62< φeff < 0.82 for intensity at the smallest wave vector. Consistent with
set A and 0.60 < φeff < 0.77 for set B. Based on the previous studies [3, 4], the scattering intensities display
rheologystudies[14], nanoemulsionsovertheserangesof a pronounced peak near q = 0.10 nm−1, corresponding
φeff behaveaselasticsolids. However,theshearmodulus to the first peak in the interparticle structure factor, as
over these ranges decreases significantly with decreasing well as additional correlation peaks at higher wave vec-
φeff,byatleastoffactorof100foreachset[14]. Efforts tor. Thestructuredisplayslittlechangewithtw overthe
to extend the ranges to even lowerφeff led to emulsions time of the experiments, consistent with the very long
that werefluid andthat haddropletdynamics thatwere shelflife ofthe nanoemulsionsamples. However,wenote
too rapid to observe in the XPCS measurements. that I(q) at the largest t in Fig. 1(a) does show some
w
deviation from I(q) at earlier t . This deviation repre-
The experiments were performed at sector 8-ID of the w
sents the largestchangein I(q) with t observedfor any
Advanced Photon Source using 7.35 keV x-rays. De- w
nanoemulsion studied.
tails regarding the beam line have been presented else-
where [15, 16]. In this study, a 20 µm × 20 µm aper- Figures 2(a) and 2(b) display results for the corre-
sponding intensity autocorrelationfunction g (q,t) mea-
ture before the sample was employed to create the par- 2
sured at a wave vector near the first peak in I(q) for
tially coherent x-ray beam. The nanoemulsions were
the nanoemulsions from set A with φ = 0.82 and set
loaded into sealable, stainless-steel sample holders with eff
B with φ = 0.60, respectively, at the same waiting
thin polyimide windows for transmission scattering and eff
timesasinFigs.1(a)and1(b). Thecorrelationfunctions
sample thickness of 0.5 mm. The loading involved ex-
decay on a time scale that increases with increasing t
tractingappropriatequantitiesofnanoemulsionfromthe w
immediately following the loading of the nanoemulsions
glass vials in which they are stored and spreading it
into the sample cells, but reach steady-state behavior at
into the sample cells with a razor. The temperature
larger t . The dynamic structure factor f(q,t) can be
of the samples was held at 25.0 C throughout the mea- w
calculated from g (q,t) using the Siegert relation [16].
surements. The scattering intensity was recorded by a 2
Modeling f(q,t) with a stretched exponential lineshape
direct-illuminated CCD area detector positioned 3.4 m
f(q,t) = f exp[(−t/τ)β], the intensity autocorrelation
beyond the sample to cover a wave-vector range of 0.05 q
nm−1 < q < 0.39 nm−1. A series of scattering images, function has the form
withatypicalexposuretimeof0.5s,wasobtainedtode- g (q,t)=1+b[f exp[−(t/τ)β]]2 (1)
2 q
termine the ensemble-averaged intensity autocorrelation
function g (q,t) over the range 3 s <t< 1000 s [15, 16]. wheref is the short-time (t< 1 s)plateauamplitude of
2 q
The minimum delay time, 3 s, was set by the sum of f(q,t),τ istheterminalrelaxationtime,β isthestretch-
the exposure time plus the time to downloadeachimage ing exponent that characterizes the shape of the corre-
from the CCD. To reduce exposure of the sample to x- lation function, and b ≃ 0.30 is the Siegert factor de-
rays, a shutter before the sample was closed during the termined from measurements on a static aerogelsample.
datatransfer. Thelongestdelaytimewaslimitedto1000 For all the nanoemulsions and all wave vectors, fits to
sto avoideffectsofmeasurementstabilitythatledto ar- Eq. (1) agree closely with the data except at very large
tificialdecaysing (q,t)atseveralthousandseconds. The t and small g (q,t). The lines in Fig. 2 are the results
2 2
measurement of g (q,t) was repeated over the course of of fits to Eq. (1) for g (q,t)>1.03, illustrating the good
2 2
3
(a) 1.25 (a)
a = 46 nm
1 φeff = 0.82 1.20 aφ = =4 60 .n8m2
eff
) )
q t 1.15
,
( q
I (
2
0.1 g 1.10
46 min
46 min
88 min
88 min
138 min 1.05
138 min
621 min
621 min
0.01
1.00
(b) 1.25 (b)
a = 36 nm
1 φeff = 0.60 1.20 aφ = =3 60 .n6m0
eff
)
q
I( 28 min q,t) 1.15 28 min
0.1 71 min (2 71 min
133 min g 1.10
133 min
184 min
184 min
245 min
1.05 245 min
427 min
427 min
0.01 588 min
588 min
1.00
0.05 0.1 0.2 0.3 0.4 10 100 1000
-1
q [nm ] t [s]
FIG. 1: X-ray scattering intensities for nanoemulsions of (a) FIG.2: Intensityautocorrelationfunctionsfornanoemulsions
dropletradius46nm(setA)withφeff =0.82and(b)droplet of (a) droplet radius 46 nm (set A) with φeff = 0.82 at q =
radius 36 nm (set B) with φeff = 0.60 measured at several 0.08 nm−1 and (b)droplet radius36 nm(set B) with φeff =
waiting times tw following the loading of the nanoemulsions 0.60 at q = 0.10 nm−1 measured several waiting times tw
into sample cells, as specified in the figure legends. The in-
following the loading of the nanoemulsions into sample cells.
tensities are normalized with respect to the intensity at the
Symbols refer to the same waiting times as in Fig. 1. The
smallestwavevector. ThepeaksinI(q)belowq=0.10nm−1
solid lines are the results of fits to Eq. (1) over the range
correspondtothefirstinterparticlestructurefactorpeak. Ad- g2(q,t)>1.03andareextrapolatedwithdashedlinesbeyond
ditional correlation peaks are also observable at higher wave thisrangetoillustrate thedeviationsfrom Eq.(1) at large t.
vector.
agreementatsmallt andacrossovertoaslowerdecayin decay in f(q,t), and slow component in which the par-
the measured g2(q,t) at larger t. ticle diffuses away from its center-of-mass position with
cooperative motion from its neighbors, leading to the fi-
nal decay of f(q,t). We thus interpret the fast partial
B. Fast Dynamics decay of f(q,t) of the nanoemulsions in terms of local-
izedmotionofthedropletswithintheconstraintsoftheir
Results for the short-time plateau value of f(q,t) ex- neighbors. A similar conclusion was reached regarding
tracted from fits to Eq. (1) are consistently less than fast dynamics in concentrated emulsions of micrometer-
one, as illustrated in Fig. 3(a), which displays values of scale droplets based on dynamic light scattering mea-
f as a function of wave vector for nanoemulsions from surements [17]. However, for both the emulsions of
q
set A with φ = 0.62 and 0.82. The observation that micrometer-scale droplets and colloidal suspensions, f
eff q
f < 1 indicates a partial decay of the dynamic struc- displays a pronounced peak at a wave vector near the
q
ture factor at inaccessibly shorttimes, suggesting a two- position of the first peak in the interparticle structure
step relaxation in the dynamics. Two-step relaxations factorS(q)[17,18]. Incontrast,f forthenanoemulsions
q
are a signature feature of glassy particle dynamics. In shown in Fig. 3(a) is largely featureless as a function of
colloidalsuspensions near the glass transition, the parti- q. Mode coupling theory, which provides a framework
cle dynamics separate into a fast localized component in for the dynamics in colloidal suspensions near the glass
which a particle moves within the restricted volume de- transition, predicts a peak in the amplitude of f near
q
fined by the positions of its neighbors, creating a partial theinterparticlestructurefactorpeak[19]. However,the
4
magnitude of the peak predicted by the theory depends
1.0
in detail on the interparticle potential and the resulting (a)
form of S(q) [18]. The relatively featureless f for the
q
concentrated nanoemulsions could thus reflect the dis-
tinct interparticle interactions between the charged, de-
0.9
formable droplets and the interparticle structure factor
that has a relatively weak correlation peak as compared
with thatofhard-sphereglasses[3, 4]. Alternatively,the fq
wave-vector independence of f could indicate that the
q
0.8
fast dynamics in the nanoemulsions are predominately
local rotational motions that potentially involve one or
more droplets.
As with wave vector, f does not show any clear vari-
q 0.06 0.08 0.10
ation with tw, consistent with an unchanging structure; q [nm-1]
however,itdoesvarysystematicallywithdropletvolume
fraction. The dashed lines in Fig. 3(a), representing the 0.95
averagesoff forthetwovolumefractions,illustratethis (b)
q
variation. Figure3(b)showstheaverageplateauvaluef¯
q
for the nanoemulsionsin sets AandB.To obtainf¯, the
q
values of f are averaged over both wave vector and t . 0.90
q w
The resulting trends, in whichthe shorttime plateauin-
| q
creases with increasing droplet volume fraction, indicate f
that the range of the localized motion becomes smaller
0.85
with increasing volume fraction, as expected for repul-
sively interacting droplets. 36 nm
46 nm
0.80
C. Slow, Non-Diffusive Dynamics 0.6 0.7 0.8
φ
eff
Whilethefast,localizedcomponentofthedropletmo-
tioninthenanoemulsionsiscommontocolloidalglasses,
FIG. 3: (a) Short-time plateau value of f(q,t) versus wave
in contrast, the slow dynamics characterized by the de-
vector for nanoemulsions with droplet radius 46 nm and vol-
cay in g (q,t) are clearly distinct from the cooperative
2 umefractionsφ =0.82(open)and0.62(filled)measuredat
eff
diffusion in colloidal suspensions near the glass transi-
tw = 130 minutes. The dashed lines show the average values
tion. Instead,theshapeofg (q,t)anditswavevectorde-
2 for each volumefraction at thiswaiting time. (b) Short-time
pendence exemplify two salient features of non-diffusive plateauvaluesaveragedoverqandtw asafunctionofdroplet
dynamicsthatareapparentlyuniversaltoarangeofdis- volumefraction for sets A (filled) and B (open).
orderedsoftsolids. First,fitstoEq.(1)giveβ =1.7±0.2,
implying a compressed exponential correlation function.
Thislineshape,whichweobserveforallofthenanoemul-
sions, displays no systematic variation with t or φ , tic velocity, v ≡1/(τq). Based on a heuristic argument
w eff 0
although some results suggest that β decreases slightly byCipellettiet al.[5,10],BouchaudandPitard[20]have
with increasing q. Second, the relaxation time τ ex- introducedamicroscopicmodelfortheseballisticdynam-
tracted from the fits varies with wave vector as τ ∼q−z ics,describingthemintermsofelasticstraindeformation
with z ≈ 1.0, as shown in Fig. 4 for the nanoemulsion in response to heterogeneous local stress. Specifically,
from set A with φ = 0.82. Similar compressed ex- Bouchaud and Pitard picture the source of this stress
eff
ponential lineshapes with τ ∼ q−1 have been observed asarandompopulationofparticlerearrangementevents
previouslyindynamiclightscatteringandXPCSstudies that create point-like dipolar stress fields whose intensi-
of colloidal gels [5, 6], clay suspensions [7, 8], dense fer- ties P(t) grow linearly over a period of time θ to a max-
rofluids [9], concentrated emulsions of micrometer-scale imum P , and they show that the resulting strain leads
0
droplets [10], and a block copolymer mesophase [11]. to a compressed exponential lineshape and τ ∼q−1 [20].
Typically, for these other disordered soft solids β ≈ 1.5, More recently, Duri and Cipelletti have shown that, at
slightly smaller than we observe for the nanoemulsions. least for dilute colloidal gels, the strain grows linearly in
Regardlessoftheprecisevalue,however,suchcompressed time only ina time-averagedsenseandthe dynamicsare
correlation functions with β >1 and τ ∼q−1 are incon- more accurately described in terms of temporally het-
sistent with purely diffusive particle motion. Instead, as erogeneous, discrete displacements [21]. An interesting
discussed previously [10], they indicate ballistic motion question is whether such temporally heterogeneous mo-
with a broad distribution of velocities and a characteris- tion also describes the strain in concentrated soft solids
5
3.0 500
a = 46 nm (a)
φ = 0.82
eff
s] 200
[ 2.5

)
τ ]
g(10 τ [s 100 a = 46 nm
lo 2.0 46 min φ = 0.62
88 min 50 eff
φ = 0.73
138 min eff
621 min φ = 0.82
eff
1.5
20
-1.4 -1.0 -0.6
-1
log (q) [nm ] 500 (b)
10

FIG. 4: Characteristic relaxation time τ as a function of a
wave vector q for a nanoemulsion with droplet radius 46 nm 200
]
(setA)andφ =0.82measuredatseveralwaitingtimes,as s
eff [
specified in the figure legend. The solid lines are the results τ a = 36 nm
forfofimts0t.8o7thtoef1o.r1m0 wτit∼hqa−nz,avwehraicghegziv=ea1n.03ex±po0n.0e9n.tz ranging 100 φφeff == 00..6606
eff
50 φ = 0.70
eff
φ = 0.77
like nanoemulsions that display this characteristic non- eff

diffusive dynamics.
20 1 00 200 1 000
A key parameter in the Bouchaud-Pitardmodel is the
time scale τq = 4πPK0γ2qθ2, where K is the elastic modulus tw [min]
of the material and γ is a friction coefficient related to
dampening of the strain motion. For t<<τq, the model FIG.5: Characteristicrelaxationtimeasafunctionofwaiting
predicts β = 1.5, as observed in most previous systems timefornanoemulsionswith(a)dropletradius46nmand(b)
displaying these non-diffusive slow dynamics. However, droplet radius 36 nm with various droplet volume fractions,
if t/τ is not very small, an effective exponent somewhat specifiedinthefigurelegends,measuredatawavevectornear
largerq than 1.5 is expected. Further, for t >> τ a new the first peak in I(q) (q = 0.08 nm−1 for (a) and q = 0.10
q nm−1 for (b)). The dashed lines are the results of fits to
regime of behavior dominates in which β = 1.25. Thus,
Eq. (3), which implies two components to the characteristic
the observed lineshapes, like those displayed in Fig. 2,
whichfollowacompressedexponentialformwithβ ≈1.7 strain velocity.
at smaller t and a more stretched decay at larger t, are
qualitatively consistent with the idea that the measure- erably among different nanoemulsions. For example, as
ments on the nanoemulsions probe a range of dynamics shown in Fig. 5(a), the nanoemulsion with φ = 0.73
eff
in which t >∼ τq. However, the considerable scatter in in set A displays no appreciable increase in τ over the
the data at very large t (where g2(q,t) <∼ 1.03 and de- measured range of tw, suggesting that the first measure-
viations from the fits to Eq. (1) in Fig. 2 are apparent) ment of g (q,t) on this nanoemulsion was performed at
2
does not permit a quantitative analysis of the lineshape toolateawaitingtimetocaptureanytransientbehavior.
in this regime. Tomodeltheevolutionofτ,wesupposethatthechar-
acteristic strain velocity has two components, one that
decays exponentially with increasing t and one that is
w
1. Relaxation Times independent of waiting time,
v =(v −v )exp(−t /Γ)+v , (2)
Figure 5(a) displays the relaxation time τ as a func- 0 i s w s
tionoft fornanoemulsionsinsetAwiththreedifferent where v is the characteristic velocity at t = 0, Γ is
w i w
dropletvolumefractions measuredata wavevectornear the decay time of the transientcomponent, and v is the
s
the first peak in I(q). For two effective volume frac- steady-state characteristic velocity reached at t >> Γ.
w
tions, φ = 0.62 and 0.82, the relaxation time grows The corresponding relaxation time is therefore given by
eff
rapidly at short tw and saturates to a steady-state value τ =(((v −v )exp(−t /Γ)+v )q)−1, (3)
at longer waiting times. Similar rapid increases in τ fol- i s w s
lowedbysteady-statebehaviorarealsoobservedwiththe The dashedlines in Figs.5(a)and5(b) arethe results of
nanoemulsions in set B, as shown in Fig. 5(b). However, fits to Eq. (3). For the nanoemulsions in which the mea-
themagnitudeofthechangeinτ atearlyt variesconsid- surements capture an appreciable increase in τ at early
w
6
tw, Eq. (3) describes the evolution accurately. The time 1000
scaleforthedecayofthetransientstrainmotionisfound 36 nm
to be in the range Γ≈10−20 minutes for all cases. For 46 nm
significantlylargerwaitingtimes,t >>Γ,thistransient
w
component contributes negligibly, and the strain motion
is dominated by the steady-state component. s]
[ 500

s
τ
2. Transient Component
The presence of two components to the strain motion
0
indicates two distinct sources of local stress relaxation.
The transient motion, which dominates at small t , has 0.6 0.7 0.8
w φ
acharacteristicvelocitythatdecaysrapidlywithincreas-
eff
ing t leading to the corresponding exponential growth
w
in τ. A similar exponential growth in τ at early tw has FIG.6: Steady-statecharacteristic relaxation time measured
been observed in a number of disordered soft solids in- atlarge waitingtimetw fornanoemulsionswith dropletradii
cluding colloidal gels [5], clay suspensions [8], and dense a = 46 nm (solid) and a = 36 nm (open) as a function of
ferrofluids [9], indicating that such an evolution in these effectivevolumefraction. Therelaxationtimescorrespondto
non-diffusive dynamics is fairly generic. This rapid evo- wave vectors near the first peak in I(q) (q = 0.08 nm−1 for
lution typically follows a significant perturbation of the a= 46 nm and q=0.10 nm−1 for a= 36 nm).
system,suchasaquenchfromfluidtodisorderedsolidor
alargemechanicalstress,andthese dynamicsthus likely
reflect a relaxationfromthe state createdby the pertur- ers [22, 23]. Thus, a waiting-time dependence essentially
bation. Thus,weassociatethetransientstrainmotionin opposite to that in Fig. 5 is observed. This contrasting
thenanoemulsionswithresidualstressthatisintroduced behavior presumably reflects both the differing mechan-
by loading the nanoemulsions into the sample cells and ical perturbation imposed in the foam studies as com-
that relaxes through slow droplet rearrangements. The paredwiththisstudyaswellasthedifferingcompliances
extent of mechanical perturbation associated with the of the two systems and suggests an analogy with the
sample loading processwasnot quantitatively controlled “over-aging” versus “rejuvenation” scenario that occurs
intheexperiments,and,therefore,theamountofresidual in the shear response of glasses [24, 25, 26, 27]. Specif-
stress introduced presumably varied among the samples. ically, the large correlation time immediately following
The wide range in the amount that τ changes at early the shearing of the foam is interpreted as a consequence
t seen in Fig. 5 likely reflects this variability. In the ofhomogenizingtheinternalstrainfield[22]. Thatis,the
w
context of stress dipoles, the perturbation thus creates shear assists the system in reaching a lower-energy con-
many dipoles at the same formation time, t = 0, and figurationmuchlikethephenomenonenvisionedin“over-
w
the subsequent dynamics are dominated by the growth aging”. In contrast, the perturbation of the nanoemul-
inintensity ofthese dipoles. We notethis scenarioisdif- sionsinvolvedinloadingtheminthesamplecellshasthe
ferent from the one consideredby Bouchaudand Pitard, opposite “rejuvenating” effect of imparting energy into
whose model assumes a steady-state formation rate ρ of the system in the form of local regions of poorly packed
new stress dipoles. (Although, the model also includes a droplets,leadingtoaninitiallydensepopulationofgrow-
mechanismforthedynamicstoevolve,or“age”,through ing stress dipoles. Future experiments onnanoemulsions
anage-dependentρthatleadstoapower-lawgrowthinτ thatinvestigatetheeffectsofshearflowoverawiderange
withage.) Onequestionthereforeiswhethertheaddition ofshearratesmightcapturethetransitionbetweenthese
of a large population of stress dipoles formed at t = 0 classesofresponse,andcouldprovidefurtherinsightinto
w
into the Bouchaud-Pitard model could capture the ex- the effects of shear on soft glassy materials.
ponentialgrowthinτ versust observedexperimentally.
w
Presumably, to obtain the observed behavior the model
would need to consider a distribution of growth periods 3. Steady-State Component
θ of these dipoles, rather than a single characteristic pe-
riod. As mentioned above,the values of τ that are indepen-
An interestingcomparisoncanbe made betweenthese dent of waiting time at large t in the nanoemulsions
w
transient droplet dynamics in concentrated nanoemul- indicate a second, steady-state source of local stress-
sions observed with XPCS and those of the bubbles relaxationevents that is distinct fromthe residualstress
of macroscopic foam studied with diffusing wave spec- of sample loading. The fact that τ is independent
troscopy(DWS). Inthe foam experiments,τ is largeim- of waiting time further suggests a means for exploring
mediately following the cessation of shear and decreases these dynamics systematically. Specifically, while such
exponentially with waiting time as the system recov- non-diffusive dynamics have been identified in a host of
7
disordered soft-solid materials, the significance of this ingitanunlikelycandidateforcreatingsourcesofstress.
strain motion for other material properties is unclear. However,wedonotethatsomeofthenanoemulsionsdid
The seeming insensitivity of the steady-state component display small variations in I(q) during the course of the
in nanoemulsions to sample history should simplify the measurements, as illustrated by Fig. 1(a), which might
study of its dependence on other material parameters, indicatethattheemulsionsaremadelessstablebythex-
and clarifying their relation to other parameters might raymeasurementitself. Thus,thesourcesofstressmight
shed lighton the role ofthese dynamics in dictating ma- originate as a consequence of the measurement. For ex-
terial behavior. As a first investigation into the nature ample, contactbetween the nanoemulsions and the sam-
of the steady-state dynamics, we have characterized the ple cell could cause droplet coalescence and such coales-
dependence of the strain motion on droplet volume frac- cence events could drive the strain response. However,
tion. Figure 6 shows the steady-state relaxation time we do not expect that the materials in contact with the
τ ≡ 1/(v q), measured for t >> Γ, as a function of nanoemulsions – specifically, stainless steel, polyimide,
s s w
φ for nanoemulsion sets A and B. Over the ranges of and epoxy – would be problematic. Also, analysis of
eff
φ investigated, the dynamics are only weakly sensi- the measurementsto discriminate dynamics alongdiffer-
eff
tive to volume fraction, with the steady-state relaxation ent wave-vector directions indicates that the dynamics
timeincreasingbyroughlyafactoroftwowithdecreasing are isotropic, suggesting that the sources of stress are
φ . As mentioned above, rheology studies have shown positioned randomly and are not restricted to the sur-
eff
that the shear modulus decreases by more than a factor faces. Radiation damage is another potential source of
of 100 with decreasing φ over these ranges [14]. local stress introduced by the measurement. However,
eff
At first glance, the relatively weak dependence of efforts to identify evidence of radiation damage by vary-
the strain motion on φ is thus surprising since the ing exposure times were negative, and the protocol of
eff
Bouchaud-Pitardmodelpredictsthatτ ∼ Kθ1/3 ,where making each measurement of g2(q,t) on a new region of
ρ2/3P0q sample should have prevented any cumulative damage.
K is an elastic constant [20]. (In the model K repre-
A future experiment that might provide insight would
sents the compression modulus, so this comparison is
be a study in which droplet coarsening is intentionally
not exact; however, a more refined model that includes
accelerated. One approach might be to investigate the
consideration of the shear modulus would differ only in
temperature dependence of the dynamics to see if the
numerical factors [20].) However, the variation in mod-
strain motion reflects the thermally activated behavior
ulus with volume fraction occurs through the increase
expected for coarsening. Another would be to employ
inthedroplet-dropletrepulsionpotentialwithincreasing
nanoemulsions comprised of oils that are less immisci-
φ , and a change in the interaction potential should
eff ble in water, so that the stability against coarsening is
also affect P , the strength of the stress dipoles driv-
0 compromised. Testingwhethersuchachangealtersqual-
ing the strain motion. If K and P possess the same
0 itatively the droplet-scale dynamics or merely increases
dependence on interaction potential, then the effects on
the characteristic velocity of the observed strain motion
the strain motion of changing φ should cancel since
eff wouldhelpelucidatethepossibleroleofcoarseninginthe
τ ∼K/P , consistentwith ourobservationthat τ varies
0 s current study.
only weakly with φ . Additional XPCS measurements
eff
on nanoemulsions with varying concentrations of NaCl Another possible source of the steady-state, non-
addedto the continuousphasesupportthis picture. The diffusivedynamicsislocal,irreversiblesheardeformation
salt,addedtonanoemulsionsfromsetAwithφeff =0.73 that results from thermal expansion in response to tem-
to concentrations as large as c=160 mM, decreases sig- perature fluctuations. In a recent microscopy study of
nificantly the shear modulus by screening the Coulom- concentrated multilamellar vesicles, Mazoyer et al. [30]
bic interaction between droplets. However, XPCS mea- observed slow dynamics corresponding to such deforma-
surementsonthenanoemulsionswithaddedsaltrevealed tion and suggested that they might be relevant to the
that the steady-state strain dynamics are essentially un- slow, non-diffusive dynamics seen in light scattering and
changed. XPCS studies of soft glassy materials. While such ir-
While the weak dependence of the non-diffusive dy- reversible shear deformation represents a physically ap-
namicsonφ andaddedsaltprovidessomeinsightinto pealing microscopic origin for the steady-state motion,
eff
the mechanisms driving the strain motion, the micro- wenotethatthetemperaturecontrolintheXPCSexper-
scopic originof the local sourcesof steady-state stress in imentslimited the fluctuationsinsampletemperatureto
thenanoemulsionsremainsaquestion. Onepossibilityis approximately±0.001K.Thisfluctuationamplitudewas
dropletcoarsening,whichisknowntodrivethedynamics significantly smaller than that in the microscopy study,
inmacroscopicfoamsobservedwithDWS[28,29]. How- raising doubt as to whether thermal expansion fluctu-
ever,thecorrelationfunctionobservedforthefoamswith ations would be large enough in the nanoemulsions to
DWS is quite distinct from the compressed exponential drive such shear deformation. An additional, related is-
lineshape measured with XPCS on the nanoemulsions, sue in the XPCSmeasurements is the possibility of local
indicatingquitedifferentmicroscopicdynamics. Further, heatingduetox-rayabsorption. Calculationsandexper-
the long shelf life of the nanoemulsions indicates that iments on materials sensitive to this effect indicate that
anycoarseningproceeds atanextremely slowrate,mak- thesampletemperaturewithinthe20µm×20µmcross
8
section of the beam increased by approximately 0.004 K general principle might underlie the formation of such
duetobeamheating. Thistemperatureincreaseoccurred sites of stress (perhaps one tied to the mechanical prop-
with the periodicity of the x-ray exposures, once every 3 erties of systems undergoing a jamming transition [31])
s, and so could conceivably play a role similar to that andcallsforfurther investigationinto the originofthese
of the temperature fluctuations in Mazoyer et al. How- dynamics and their role in determining material proper-
ever,sincetheamplitudeoftheincreasewassosmall,we ties. The relativelyunique aspectof the dynamics in the
believe the thermal expansioncaused by this fluctuation nanoemulsions, which is also shared by the recently re-
again was likely to be too small to drive the observed ported dynamics in a polymer-based sponge phase [11],
dynamics. A future experiment that might elucidate the istheirsteady-statebehavior. Thisapparentlackofevo-
potential role of thermal fluctuations would be to intro- lutioninthestressrelaxationhasanumberofsignificant
duce temperature oscillations of varying amplitude and implications. First, it highlights an observation made
frequency to test whether they systematically influence previously [6] that these dynamics are distinct from tra-
the steady-state non-diffusive dynamics. ditional aging seen below the glass transition in hard
disordered solids such as molecular liquids and polymer
melts[32,33,34]. Second,itopensthepossibilityforsys-
IV. CONCLUSION tematically tuning the strain motion by changing mate-
rialparameters. Thislatterfeaturemakesthenanoemul-
In conclusion, we have found that the slow dynam- sions a particularly appealing model system for future
ics of nanoemulsions probed with XPCS are character- investigationsintothenatureoftheseslow,non-diffusive
ized by non-diffusive motion similar to that observed in dynamics.
anumberofdisorderedsoftsolids. Thecompressedexpo-
nential form for f(q,t) and inverse relationship between
relaxation time and wave vector that are the signatures
of these dynamics match well to the predictions of the Acknowledgements: We thankR.Bandyopadhyay,D.
model introduced by Cipelletti et al. [5, 10] and devel- Durian, and S. Mochrie for helpful discussions and S.
oped by Bouchaud and Pitard [20] in which this motion Narayanan and A. Sandy for their assistance. Funding
corresponds to strain from heterogeneous, local stress. was provided by the NSF (DMR-0134377). Use of the
However,thelargerangeofdisorderedsoftsolidsinwhich APS was supported by the DOE, Office of Basic Energy
these dynamics havebeenobservedsuggeststhat amore Sciences, under Contract No. W-31-109-Eng-38.
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