Direct measurement of ballistic to diffusive crossover in colloidal particles in simple and complex fluids

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by  Andrew P. Hammond

Direct measurement of ballistic to diffusive crossover in colloidal particles in simple
and complex fluids
Andrew P. Hammond, Eric I. Corwin
Materials Science Institute and Department of Physics, University of Oregon, Eugene, Oregon 97403
(Dated: January 26, 2017)
Athermalcolloidsuspendedinaliquidexperiencestwotypesofmotion. Atshorttimescalesthe
motion will be ballistic and at long time scales diffusive. We directly observe a free floating tracer
particle’smotioninbothregimesanditstransitionbetweenthetwo. Byanalyzingthemotionofthe
tracerparticleinadenseNewtonianfluid,wemeasurethetemperature,viscosity,andtracerradius.
In addition, the full transition gives us added insight into the structure of the fluid. We confirm
this by examining a tracer’s motion in a non-Newtonian Maxwell fluid. We find a significant shift
7 in the transition between ballistic and diffusive motion with the addition of intermediary thermal
1 springlikemotion. TheMaxwelltransitionalsoshowsanincreaseintheeffectivemassofthetracer
0 particle.
2

n
a At very short time and length scales the diffusive mo- at a frame rate of 40,000 frames per second, yielding a
J tion of a Brownian particle breaks down into a series of minimum lag time between measurements of 25 µs. We

4 individual ballistic flights. At a time when it was impos- track the particle’s motion with nanometer accuracy as
2 sible to contemplate the direct imaging or observation of shown in figure 1.

individual molecules, this idea was used by Einstein as A rigorous theoretical framework connecting fluid
]
t definitive proof for the molecular composition of fluids propertiestotheobservableMSDofacolloidisprovided
f
o [1]. Indeed,themicroscopictimeandlengthscalesforthe by solving the Langevin equation of state. The simplest
s ballistic motion are so small that direct measurements model for a fluid is the ideal gas [15, 16] whose Langevin
.
t have only recently become possible [2–5]. These experi- equation has only two terms. The MSD for an ideal gas
a
m ments have used optical traps to confine a test particle hastwoasymptoticbehaviors,atlongtimesitisdiffusive
within a harmonic well, allowing a high precision mea- (MSD∝t)andatshorttimesballistic(MSD∝t2). The
-
d surementoftheshorttimemotionbutatthecostofaloss transition between these two behaviors is exponential,
n of information about the crossover to diffusive behavior. and thus rapid.
o
Herewepresentdirectmeasurementsofthefulltransition Amoresophisticatedmodelfordensefluids,motivated
c
[ between ballistic and diffusive motion for a freely mov- by early computer simulations [17, 18], is achieved by
ing colloid suspended in simple Newtonian and complex addinganeffectivemasstermandamemorytermtothe
1
Maxwell fluids. These measurements are achieved using idealgasmodel[7,9]. Theeffectivemasstermmodelsthe
v
a high speed camera, intense illumination, and an ac- frictionally bound fluid that is attached to the particle
0
7 curate tracking algorithm [6]. These measurements have and the memory term models the inertial interaction of
0 sufficientprecisiontounambiguouslydistinguishbetween theparticlewithnearbymovingfluid[19]. Atsufficiently
7 microscopic models for fluids [7–14]. In a simple Newto- short timescales and close to the speed of sound in the
0
nian fluid we find that nearby fluid is frictionally bound fluid this model breaks down and is replaced with the
.
1 totheparticle,increasingitseffectivemass. Bycontrast, simple ideal gas model.
0 in a complex Maxwell fluid made up of interacting mi- ThesemodificationstotheLangevinequationwerean-
7
celles the region of bound fluid is much greater leading alyticallysolved[10]undertheassumptionsthatthefluid
1
: to an even larger effective mass. The functional form of is viscous and incompressible, the Reynolds number is
v the full transition provides a direct measurement of the low, and the test particle is a hard sphere [8, 20]. The
i
X temperature and rheological response of the fluid. predicted MSD is included in the supplementary infor-
r In this experiment we directly measure the mean mation. The dense fluid MSD differs from the ideal gas
a
squared displacement (MSD) of a freely floating colloid MSDintwosalientways: 1)Itgivesrisetoaslowerballis-
insimpleandcomplexfluids. Weexamineafreefloating tic velocity, caused by the increased effective mass of the
unperturbed polystyrene test particle with a radius of particle. 2) It has a much gentler crossover between bal-
21.8µm(Spherotech,ppx-400-10). Weplacetheparticle listicanddiffusivemotion,causedbytheinertialmemory
in either density matched salt water, a Newtonian fluid, of the liquid.
or a density matched Cetyltrimethylammonium chloride Non-Newtonian fluids, however, can have much more
(CTAC) and water solution, a non-Newtonian Maxwell complicatedLangevinequations[11,12]. Oneofthesim-
fluid. Our fluid sample is isolated from external influ- plest non-Newtonian fluids is a Maxwell fluid, character-
ences inside of a silicon well sandwiched between a glass ized by a single relaxation time between spring like and
slide and cover slip. Using a high speed camera (Vision viscous like behavior. This thus requires the addition of
Research Miro 310) we image the motion of the particle a decaying spring term to the Langevin equation. This
2
it was placed in a Fastwell silicon spacer cavity between
a slide and a cover slip. The chamber was sealed with
vacuum grease to ensure that there were no air bubbles.
The silicon spacer had a width of 2.4 mm and circular
void with a radius of 5 mm allowing the colloids to be
imaged far from wall effects. The slides were cleaned
withpiranhasolutionanddriedwithnitrogengas,which
0.4 μm
removed any coatings on the slides.
For the Maxwell fluid, we added the same polystyrene
beads to CTAC at a concentration of 2.5×10−3 % w/v.
The fluid density was measured to be 1.055×103 kg/m3
slightly lower than the average density of the beads.
However, the density is close enough that beads did not
fall out of suspension until well after the measurements
were complete. When preparing samples, we used a pro-
cessalmostidenticaltotheoneforwater. Themajordif-
ference was that the sample was not sonicated or mixed
ahead of being added to the Fastwell because the fluid
acted solid-like when exposed to high frequency agita-
10 μm
tion. Instead, the sample was slowly mixed using a low
frequency mixer.
FIG. 1. A single colloid suspended in water viewed through All data was collected on a Nikon TE2000s micro-
the microscope. The path the particle travels for the next 2
scope on a floating stage optical table in a climate con-
seconds is shown in blue.
trolledroom. Illuminationwasprovidedbya500mWred
LED (Thorlabs LED635L) shining through the micro-
scopecondenser. InbetweentheLEDandsampleaneu-
additional term makes a closed form MSD difficult to
tral density filter on a swivel mount was added to allow
solve for, but we expect the addition of an intermediate
initial setup to be done without excessive local heating
plateauregimeintheMSDcorrespondingtothebehavior
of the sample. The sample was encased in a small card-
of a thermal spring.
board box for isolation from acoustic vibrations. Images
In this experiment, we used polystyrene hard spheres weregatheredthrougha50xlens(NikonLUplanELWD
with a radius of 21.8 µm as our tracer particle. 50x/0.55 B inf/0 WD 10.1) using a Phantom M310 high
Polystyrene was chosen because it is easily density speed camera. Videos were taken at 40,000 fps (T = 25
matched to water using NaCl with only minimal, and µs) with an image size of 192x192 pixels and a magnifi-
known, changes to the viscosity. The size of the particle, cation of 0.4µm/pix. When filming a particle all motor-
much larger than those in optical trap experiments, was izedelementsonthemicroscopeandcamerawereturned
chosenbecausethetrackingprecisionaswellastheideal off to eliminate small vibrations. Once a particle was
gas transition time and length increase with increasing found, filming lasted 2.84 s (113,600 frames) after which
radius. Wechosetousewaterasourexperimentalliquid the LED was immediately shut off and the ND filter re-
because of its ubiquity in experiments and its relatively placed.
low viscosity. For our setup, the Reynolds number is We used a radial center tracking algorithm[6] to find
2.4×10−9. This system fulfills all of the underlying as- the center of the colloids in progressive frames of the
sumptions required by the dense fluid equation. When video. Using a combination of simulations and tracking
selecting a Maxwell fluid we chose to use a solution of testparticleswhichwerestucktotheslide,wefoundthat
Cetyltrimethylammonium chloride (CTAC) mixed with the algorithm did not exhibit a preferred direction. We
water. ThismixturehasbeenfoundtoexhibitaMaxwell found that the mean position error was about 1.5 nm in
fluid behavior caused by worm-like micelles [12, 21] and each frame.
is a commonly studied Maxwell fluid. Fortunately, at We calculate the MSD from our measured data as
the concentrations where a Maxwell behavior is exhib- MSD(τ) = (cid:104)(cid:107)(cid:126)x(t+τ)−(cid:126)x(t)(cid:107)2(cid:105) where (cid:126)x(t) is the mea-
ited, the solution has a density range which overlapped sured position of the particle at time t, τ is the lag-time
the tracer particle density. between position measurements, and angle brackets de-
Test particles were placed into a deionized water mix- note a time average. The MSD for a representative par-
tureat5×10−3%w/vofcolloid. Thecolloidsareslightly ticle is plotted as green squares in figure 2. This MSD
denser than water, so NaCl was added to density match exhibits a small drift at long times, past about 0.1 s,
the system at a measured value of 1.06×103 kg/m3. Af- and a noise floor at very short times. The drift is likely
ter sonicating and degassing the colloid-water solution, theresultofconvectiveflowswithinoursamplechamber,
3
shown in blue in figure 2. The dense fluid model, how-
-13
10 ever, fits exceedingly well over the entire range of mea-
sured lag-times. The model depends on four physical
10-14 parameters: 1) temperature, 2) particle radius, 3) fluid
density,and4)fluidviscosity. Ofthese,weindependently
-15 measure the fluid density prior to observation. The fluid
10
) viscosityofsaltwaterisaknownfunctionofdensityand
2
m
the temperature [22]. Therefore, we have only two in-
( -16
ar 10 dependent fitting parameters: temperature and particle
V 0.6
D, -17 radius. To this, we add a third fitting parameter to de-
MS10 bility0.4 scrWibeeitnhdeepmeangdneinttuldyefiotf1t8hemneoaissuereflmooern.tsusing18differ-
a
-18 b ent particles, shown in figure 3. On average, the particle
10 Pro0.2 radius was found to be 20.5±0.8 µm, within tolerance
-19 of the manufacturer’s quoted radius. The average tem-
10 0
-4 -2 0 2 4 peraturemeasuredbyourfittingwasfoundtobeslightly
Distance (nm)
higher (297±4.5 K) than the measured room tempera-
-4 -3 -2 -1
10 10 10 10 ture (293±2 K), likely the result of local heating from
Time (s) the intense illumination. The noise floors for the mea-
surement were found to range from 1.2×10−18 m2 to
FIG.2. Thevarianceofthecolloid’strace,showninfigure1. 2.4×10−18 m2. Thus fit, the dense fluid functional form
Thegreensquaresshowthemeansquareddisplacement. The is indistinguishable fromthe dataovermuchof ourmea-
blackdotswiththeerrorbarsshowthevariancewiththenoise sured range. To characterize the agreement, we plot the
floor subtracted, in this case 2.34×10−18 m2. In red is the
residual percentages, and find them to be unbiased and
dense fluid MSD fit with Temperature 302 K, radius 21 µm.
with error less than 5% over at least two decades of lag
Forcomparison,theidealgaspredictionisplottedinbluewith
time, as shown in figure 3. At longer times, where drift
diffusiveasymptotematchingthedata. Theidealgasusesthe
same fitting parameters as the dense fluid fit. We propagate and sampling errors increase, the percent error increases
the localization error of the position measurement as well as as well.
samplingerrorthroughourcalculationtoobtainerrorbarsfor WeperformsimilarexperimentsinaMaxwellfluidcre-
theplotasdescribedinthesupplementarymaterial. Inset: A ated with a solution of CTAC and water as described in
histogram of measured positions for a stranded particle.
the methods section. This solution has been found to
exhibit viscoelastic behavior characteristic of worm-like
micelles[12,21]. Followingtheprocedureoutlinedinthe
driven, perhaps, by local heating of the sample. How- methods section, we calculated an MSD for a particle
ever, this drift can be easily removed by calculating the suspended in this fluid, shown with green squares for a
variance of particle position as a function of lag time as representativetrialinfigure4. Asinthecasewithwater,
Var(τ)=MSD(τ)−(cid:107)(cid:104)(cid:126)x(t+τ)−(cid:126)x(t)(cid:105)2(cid:107). Thenoiseflooris we see a minimum noise floor at short times and a long
caused by photon shot noise in our camera contributing timedrift. Thedriftinthemeasurementcanberemoved
to uncertainty in the localization of a particle. We can byusingthevarianceasdescribedabove. Thenoisefloor
directly measure this noise by tracking a particle fixed can also be estimated and subtracted to calculate a true
to a slide and find it to be independent and identically variance for a particle within a CTAC fluid.
distributed Gaussian noise with a variance of approxi- The plotted variance for a Maxwell fluid, of which a
mately 2 × 10−18 m2 (inset to figure 2). The precise characteristicexampleisshowninfigure4, hasthreeno-
value of this noise variance changes from run to run due table features. First, at short times the motion is clearly
to variations in particle size and particle focus (due to ballistic. Thebestfitprefactorfortheasymptoteishow-
changesinz-position). Becausethisnoiseisindependent ever considerably lower than the one predicted by either
and identically distributed for a given measurement we the ideal gas or dense fluid models, corresponding to an
can simply subtract the noise floor from our measure- effectivemassfourtimeslargerthantheparticlesmassor
ment to find the true variance of our particle, plotted as anentrainedregionofradius33µm. Thisincreasedeffec-
black circles in figure 2. tive size of the particle can be understood using the fact
The plotted variance clearly shows a ballistic regime that the surrounding fluid contains worm-like micelles.
below about 10−3 to 10−4 s, a crossover regime up till The micelles in contact with the test particle effectively
about 10−2 s, and a diffusive regime for longer times. formtentaclesstretchingoffintothefluid,thusextending
As expected, when plotted against the ideal gas model the effective radius. Second, the variance shows a clear
matching the diffusive asymptote it is immediately clear secondaryplateauwhichisindependentofthenoisefloor.
that it does not describe the behavior of our system, This plateau is characteristic of thermally damped mo-
4
10-13
Temperature (K)
310
10-14 305 -17
10
300
10-15 295 )
2
m
2m)10-16 228900 Var (10-18
ar ( 47 Test Particle Diameter ((cid:541)m) D,
V10-17 45 MS
-19
Noise Floor (nm2) 43 10
10-18 3
41
2
10-19 1 39
-20
0 37 10 -4 -3 -2 -1
10 10 10 10 10
Time (s)
5
or
Err 0 FIG.4. TheMSDandadjustedvarianceforaparticlemoving
% in a Maxwell fluid. The green squares show the unadjusted
-5
MSD.Theblackdotsshowthenoisefloorsubtractedvariance.
-10 In this figure the noise floor used is 2.2883×10−18 m2. The
10-4 10-3 10-2 10-1 blue, red and magenta lines are the ballistic asymptotes cor-
Time (s) responding to the ideal gas, dense fluid, and fit respectively.
FIG. 3. Top: The noise floor adjusted variance for 18 differ-
entvideos. Inset: Thefittedtemperature,testparticleradius,
andnoisefloorforthedifferenttrials. Theblacklineshowsthe laser trap [24]. As such, high speed single particle track-
meanvalueswithagreystandarddeviation. Bottom: Resid- ing promises to become an important tool in the study
ual percentages showing by what percent the measurement
of the fundamental structure of liquids.
deviatesfromtheindividualfitsforeachofthecorresponding
variance’s shown above. We thank Raghu Parthasarathy, Tristan Ursell, and
MikeTaorminaforhelpfuldiscussions. Wealsothankthe
University of Oregon machine and electrical shop staff.
tionconsistentwithaMaxwellfluid’spredictedbehavior ThisworkwassupportedbyNationalScienceFoundation
forhighfrequencies. Finally,forlongtimes,neartheend (NSF) Career Award DMR-1255370.
ofthevariancethereismovementawayfromtheplateau
indicative of the beginning of the transition to diffusive
motion.
In this experiment, we resolve the functional form of
[1] A. Einstein, Ann. Phys. 322, 549 (1905).
the ballistic-diffusive crossover which reveals the funda-
[2] B. Luki, S. Jeney, C. Tischer, A. J. Kulik, L. Forr, and
mental length and time scales between individual and
E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
collective interactions in both Newtonian and Maxwell [3] R. Huang, I. Chavez, K. M. Taute, B. Luki, S. Jeney,
fluids. In so doing we have created a microscale first- M.G.Raizen, andE.-L.Florin,NatPhys7,576(2011).
principles thermometer based on the kinetic theory def- [4] T. Franosch, M. Grimm, M. Belushkin, F. M. Mor,
inition of temperature. This method allows for extract- G. Foffi, L. Forr, and S. Jeney, Nature 478, 85 (2011).
[5] S.Kheifets,A.Simha,K.Melin,T.Li, andM.G.Raizen,
ing rheological information at extremely high frequen-
Science 343, 1493 (2014).
cies from single particle motion, as it is performed at a
[6] R. Parthasarathy, Nat Meth 9, 724 (2012).
microscopic scale with correspondingly high spatial res-
[7] R. Zwanzig and M. Bixon, Phys. Rev. A 2, 2005 (1970).
olution whereas rheometers are necessarily macroscale. [8] A. Widom, Phys. Rev. A 3, 1394 (1971).
These results provide an independent method for testing [9] E. J. Hinch, Journal of Fluid Mechanics 72, 499 (1975).
the models for the microscopic structure of dense fluids. [10] H. J. H. Clercx and P. P. J. M. Schram, Phys. Rev. A
This method demonstrates the ability to measure mul- 46, 1942 (1992).
[11] J. H. van Zanten and K. P. Rufener, Phys. Rev. E 62,
tiple transitions between dominant effects in viscoelastic
5389 (2000).
materials, an area where laser traps have difficulty be-
[12] C. Wilhelm, J. Browaeys, A. Ponton, and J.-C. Bacri,
cause of the effects of confinement [23]. This method is
Phys. Rev. E 67, 011504 (2003).
also able to examine the influence of long range interac- [13] M. Atakhorrami, D. Mizuno, G. H. Koenderink, T. B.
tions, such as wall effects, without interference from the Liverpool,F.C.MacKintosh, andC.F.Schmidt,Phys.
5
Rev. E 77, 061508 (2008). 69, 21 (1975).
[14] Y. L. Raikher, V. V. Rusakov, and R. Perzynski, Soft [20] L. D. Landau & E.M. Lifshitz, Fluid Mechanics (Perga-
Matter 9, 10857 (2013). mon Press, 96-98, 1959).
[15] D. S. Lemons and A. Gythiel, American Journal of [21] R.Oda,J.Narayanan,P.A.Hassan,C.Manohar,R.A.
Physics 65, 1079 (1997). Salkar, F. Kern, and S. J. Candau, Langmuir 14, 4364
[16] G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1998).
(1930). [22] M. H. Sharqawy, J. H. L. V, and S. M. Zubair, Desali-
[17] A. Rahman, The Journal of Chemical Physics 45, 2585 nation and Water Treatment 16, 354 (2010).
(1966). [23] M. Grimm, S. Jeney, and T. Franosch, Soft Matter 7,
[18] B. J. Alder and T. E. Wainwright, Phys. Rev. A 1, 18 2076 (2011).
(1970). [24] B. U. Felderhof, The Journal of Chemical Physics 123,
[19] R. Zwanzig and M. Bixon, Journal of Fluid Mechanics 184903 (2005).