Linear and Nonlinear Microrheology of Dense ColloidalSuspensions

Laurence Wilson1, Rut Besseling1 Jochen Arlt2 and Wilson C K Poon1

(1) COSMICSchool of Physics, The University of Edinburgh (as part of SUPA),

James Clark Maxwell Building, Mayfield Road, Edinburgh, EH9 3JZ, Scotland

(2) Department of Physics, The University of Reading,JJ Thomson Physical Laboratory, Whiteknights, PO Box 220, Reading, RG6 6AF UK

ABSTRACT

The length and time scales accessible to optical tweezers make them an ideal tool for the examination of colloidalsystems. Embedded high-refractive-index tracer particles in an index-matched hard sphere suspension provide‘handles’ within the system to investigate the mechanical behaviour. Passive observations of the motion of asingle probe particle give information about the linear response behaviour of the system, which can be linkedto the macroscopic frequrency-dependent viscous and elastic moduli of the suspension. Separate ‘dragging’experiments allow observation of a sample’s nonlinear response to an applied stress on a particle-by particlebasis. Optical force measurements have given new data about the dynamics of phase transitions and particleinteractions; an example in this study is the transition from liquid-like to solid-like behaviour, and the emergenceof a yield stress and other effects attributable to nearest-neighbour caging effects. The forces needed to breaksuch cages and the frequency of these cage breakign events are investigated in detail for systems close to theglass transition.

Keywords: laser trapping, optical tweezers, trapping potential, colloidal glasses, microrheology

1. INTRODUCTION

The term ‘colloidal system’ describes a family of substances characterised as a dispersion of finely divided particlesin a chemically distinct suspension medium. The suspended particles are small enough for Brownian motion toinfluence their dynamics, yet too large for quantum physics to play an appreciable role. Colloidal systems are ofinterest for many reasons; as well as the research motivated by industrial needs (colloid science has applicationsin areas ranging from the preservation of foodstuffs to drilling mud removal), a carefully designed colloidalsuspension may serve as a model thermodynamic system. The simplest form of colloidal substance is a so-calledhard sphere (HS) suspension - one in which the interaction between particles is limited to hydrodynamic couplingand a billiard-ball-like interaction when particles collide. This basic system may be modified to mimic a variety ofatomic phenomena; one particular success has been the use of polymers to induce a depletion attraction betweencolloidal particles, leading to novel phase behaviour.1

In our experiments, particle concentration, expressed as a volume fraction (the fraction of sample volumetaken up by the particles divided by the total volume) is the only parameter that is varied. The hard-spheresystem that is the subject of this study has a much simpler phase behaviour than that of the attractive-particlecase mentioned above. The phase diagram shown in fig.1 shows the equilibrium phase state for a colloidalsuspension at a particular volume fraction. One particular region of interest here is the region between φ ≈ 58%and φ ≈ 64% where the equilibrium state is crystalline (φ ≈ 64% is the random close packing (RCP) limit, whereparticles are largely immobile, even on a local scale). In this region a metastable amorphous ‘glassy’ state maybe achieved with a structure that persists for very long times, often in terms of years.

Further author information: Corresponding author:L. Wilson, E-mail: Laurence.Wilson@ed.ac.uk, Telephone: +44 (0)131 6505121

Optical Trapping and Optical Micromanipulation III, edited by Kishan Dholakia, Gabriel C. Spalding,Proc. of SPIE Vol. 6326, 63261P, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.681569

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F+C Crystal

asSeS

Fluid

ib 20 30 40 50—

60 70 30 (°o)RCP CP

In the glassy state each particle is prevented from diffusing long distances (except at very long timescales)by a cage of nearest neighbors but is free to diffuse over very short lengthscales (l � a where a is the particleradius). The resultant structure is distinct from a colloidal crystal in that although it is largely frozen, there is nolong-range order. The structure and dynamics of colloidal glasses have been extensively studied using confocalmicroscopy,2 and macrorheolological studies3 have also yielded information about the bulk behaviour of glassesunder oscillatory strains.

Figure 1. Hard sphere phase diagram

Optical tweezers4 are ideally suited to making microcrheological measurements.5 Use of a carefully refractiveindex-matched suspension combined with suitable tracer particles allows the experimenter to reach inside theglass and tug at individual particles, while measuring the reaction. The force measurement system here uses aquadrant photodiode in the back focal plane of the condenser; the properties of this type of system have beenextensively described and characterised elsewhere.6, 7

2. MATERIALS AND METHODS

The behaviour and characterisation of a colloidal systems is simplified by using neutrally buoyant particles.As the particle density is essentially fixed (subject to solvent absorption and consequent swelling - see below),varying the composition is the only option to change the relative densities. As a result, a mixture of organicsolvents is used: decahydronaphthalene, which goes by the trade name of ‘decalin’, and is less dense than bulkPMMA; and cycloheptyl bromide (CHB), which is denser than bulk PMMA. Both substances were also chosenfor their refractive indices, which lie either side of the refractive index for bulk PMMA (see below). The particleswere originally received suspended in decalin; the suspension was centrifuged so as to form a sediment and‘supernatent’ liquid which was decanted. CHB was added to the sediment (a ratio of about 4:1 CHB to decalininitially), which was re-rispersed and then centrifuged at high speed and room temperature, so as to ascertainthe quality of the density matching. More of the appropriate solvent was added to the resulting sediment orcream, and the whole sample was redispersed by violent agitation and then centrifuged again. This process wasrepeated until the suspension neither creamed or sedimented under centrifuge. A sample is only density matchedacross a small range of temperatures; this means that a sediment may be formed by centrifuging a suspension ata temperature higher than its neutrally buoyant range. If the supernatent fluid above the sediment is decanted, asample volume of known weight and an assumed volume fraction of φ ≈ 64% is obtained (and as the particles areneutrally buoyant at room temperature, weight and volume fraction become interchangeable). Samples below thisvolume fraction are then obtained by dilution; as the particles are fluorescently tagged, the volume fractions canbe checked by confocal microscopy. A complication in the characterisation of PMMA samples is that the particlesare known to swell in some solvents, but a useful side effect of the lengthy density-matching process is that as it

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iteratively approaches the correct solvent composition, allowing the particles to arrive at their equilibrium radiusas it does so. The solvents used, particularly CHB, induce charges on the particles’ surfaces. This lessens theapplicability of the hard-sphere approximation, by introducing a long-range interaction between the particles.Literature sources8 have reported that in the case of the particles and solvents here, the charge-repulsion can bemitigated by the addition of a salt into the solvent (tetrabutylammonium chloride). The appropriate amount ofthe salt was added to the sample in order to lessen the charge-repulsion effect.

Refractive index matching is also necessary, to minimize the affect of the laser trap on the host particles.The refractive indices of the solvents were measured using a white-light Abbe refractometer at 22◦C and foundto be nCHB = 1.50 and ndeca = 1.47, placed either side of the literature value for bulk PMMA (at the SodiumD line) nPMMA = 1.49. The refractive index of a bulk material often differs from that of micron-scale particlesmade from the same material, and not necessarily in a predictable way. As a result, the best way to ensure goodrefractive index matching between host particles and tracers is to make up a suspension at a known refractiveindex, and then view the suspension under illumination at the tweezing wavelength (λ = 1064nm in this case).The effect of adding more of one component of the solvent mixture can then be compared between samples.Other sources9 have shown that the optical gradient force is proportional to a factor describing the difference inrefractive index between the trapped particle and the immersion medium:

F ∝(

np

nm

)2

− 1,

where np is the refractive index of the particle and nm that of the suspension medium. The measurements ofrefractive index taken using a white-light refractometer allow the tuning of solvent refractive index to a precisionof ±0.0003, which (although not wavelength-specific) offers good discrimination between solvent compositions.The imaging of particles in solvents whose refractive indices lie ±0.001 either side of the value 1.497 arrived atdue to density matching show little obvious difference in the 1064nm system. As a result, the density matchingsolvent was chosen as the final suspension medium.

The sample cell used was constructed from a standard glass microscope slide cut in half, then bridged usingtwo coverslips, which were glued in place using UV curing glue. The resulting chamber was around 15×15×1mmin size. One end of the chamber was then completely sealed using UV curing glue while the other end was partiallysealed, leaving a hole about 1mm wide by 5mm long to allow the colloidal sample to be loaded into the cell. Asmall piece of iron wire, about 1mm long and 300µm in diameter was cleaned with ethanol, dried and droppedinto the sample space10; this was moved around the chamber using a magnet about twice per day so as to breakup crystallites which may have nucleated in the suspension. Finally, the cell was plugged with a suitably-sizedpiece of glass which was glued into place with epoxy resin.

The sample was mounted on a force measurement optical tweezers setup which was designed and built alongsimilar lines to the apparatus in other work6 incorporating a microscope constructed from ’off the shelf’ opticalcomponents [fig 2]. The laser light forward-scattered by the tracer particle in the suspension was collected bya high numerical aperture oil-immersion condenser, the back focal plane of which was imaged on to a quadrantphotodiode. The voltages across the quadrants of the photodiode were recorded at a sampling rate of 20kHzby a National Instruments analog data acquisition card. The microscope stage was built with a piezoelectrictransducer (PZT) whose axis of extension was parallel (in the optical beam path) to one of the photodiode’s axes.Measurements of the displacement of a particle being dragged throught a pure solvent medium (i.e. a Newtonianfluid) have been analysed, as has the x-y correlation. The crosstalk between the two directions perpendicular tothe laser beam propagation direction is limited to below five percent on average, which is a value close to theminimal reported value elsewhere.11 The first set of experiments were preliminary calibration measurements,in order to ascertain trap stiffness and to calibrate the position sensitivity of the back focal plane measurementsystem. The calibration is performed in a sample with no host particles, just tracers in a bare Newtonian fluid.This gives a ‘baseline’ reading, which can be compared to the case of suspensions

Initially, calibration measurements were performed, examining the motion of a tracer bead in pure solventmedium. These were performed in order to provide calibration data for the later force measurements, and to givea consistency check to make sure that the results obtained were accurate. Initially, a tracer particle was held at aconstant position, with the trap focus 20µm above the lower surface of the sample cell. The displacements of the

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BeamExpander

Dichroic QPD(QuadrantPhotodiode(

Mirror

—Aperture•.____— Oil Immersion

Condenser

Ir Oil ImmersionObjective

Laser(?1O64nm(

2J2 PBS

BeamDump

Sample

Aperture BeamRelay

DichroicMirror

CCD

Figure 2. Optical setup

far-field scattering intensities are recorded and the power spectrum of the particle movements examined. Thesolvent viscosity (2.56cp) was determined by rheometry (using a concentric cylinder geometry), while the radiusof the probe particle was determined by scanning electron microscopy performed by Microparticles GmbH (whosupplied the melamine probe particles). Knowing these values, and assuming a harmonic trapping potential, afit of the data yields values for the sensitivity of the system (i.e. converting volts to nanometers) and the trapstiffness (converting nanometers to piconewtons).

The initial measurement in the colloidal samples were made using a tracer particle in a static optical trap,suspended in the host particles. The Brownian motion of the particle in a harmonic potential is recorded, andcompared with that of the same type of tracer in bare solvent. Finally, the trapped particle was held whilethe stage and host particles were translated around it [figure 3], by means of the PZT. This has the same neteffect as dragging the tracer through the suspension; this effect is compared with simulations performed byour collaborators.12 Macrorheological experiments3 have found that the mechanical properties of a suspensionchange rapidly around the glass transition (althought there is still some debate as to exactly where this occurs);elastic behaviour and yield stresses emerge, as well as shear thinning behaviour and flow at higher shear strains.These experiments go some way towards constructing a microscopic description of the same phenomena, takinginto account the dynamics of the tens and hundreds of particles in the microscopic context as opposed to around1018 involved in macrorheological experiments.

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T

Tracer

Hos

Host

Figure 3. Notional movement of particle during the final part of the experiment; host suspension and solvent moverelative to the tracer

Experiments were conducted by translating the stage at a constant velocity, and recording the displacementsfrom equilibrium of the tracer particle. These could then be analysed to reveal the average cage strength withina glass (from which a yield stress can be extracted), as well as qualitative information about the nature of cagebreaking events (see results section).

3. RESULTS

3.1. Calibration and Static (Linear) Measurements

Comparison of zero volume fraction (bare solvent) power spectra with higher volume fraction results show severalqualitative differences. Firstly, the high-frequency viscosity increases with increasing volume fraction, by factorsof 1.8 and 2.2 for the 10% and 40% samples respectively. This is due to hydrodynamic interactions betweenthe tracer and the host particles, which also lowers the corner frequency accordingly. The underlying physicaldifference between the bare solvent and the sample with host particles is the transition from Newtonian fluidbehaviour to that of a viscoelastic medium. The relaxation timescale for micron-sized particles is much greaterthan that for fluid molecules, and appears in our experiments as a solvent ‘memory’; the viscosity experiencedis a function of the particle’s history. If one assumes the generalized Stokes Einstein behaviour for a particle ina viscoelastic medium

γ̃ = 6πaη̃(ω),

the power spectrum of the probe motion is given by:

|x̃(ω)|2 =2kT

6πaω

−G′′(ω)|κ/(6πa) + G∗(ω)|2

where G* is the complex shear modulus and G” is the viscous loss modulus - the Kramers-Kronig relations maybe used to extract the real part of G* (the storage modulus). The experimental results of others13 have shownthat G* is a very weak function of frequency at low frequencies; as expected, this becomes more obvious at highervolume fractions [fig.4]. The fits in the graph were obtained using a G” with a frequency dependence of ω0.3 asseen in (13). At even higher volume fractions (approaching and crossing the glass transition - not shown), thelow-frequency portion of the power spectrum tends to 1/ω behaviour, which can be explained by the flattenedfrequency dependence of G”.

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ll [2]120

fli2bj2ccmdU IL0W drnjipunuj 2 TSCPCE OLLIm

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

0.01 0.1 1 10 100 1000

norm

alis

ed P

SD

Frequency [Hz]

Static power spectrum at different volume fractions

Bare solventφ = 10%φ = 40%

Figure 4. Comparison of power spectra at different volume fractions

3.2. Dragging (Nonlinear) Measurements

The results of simulations12 have shown that when a particle is dragged through a dense (φ >∼ 58%suspension,a ‘wake’ is formed that persists for relatively long times; this means that if a particle is dragged back througha region that it has already traversed, it experiences a momentary decrease in the apparent viscosity of thesuspension. Figure 5 shows a section of a dragging measurement where the particle is moved backward andforward through the same region of a sample. Some evidence of this effect has been seen [fig.5]; the regionbetween 80 and 83 seconds (equivalent to a distance of 10µm, compared to the particle radius of 0.85µm) showsa region of lower effective viscosity than the particle experiences when it has been dragged for longer times. Thisbehaviour is also ahown in the region between 52 and 54 seconds in the same figure.

Figure 5. Example of suspension recovery after perturbation - drag speed 3.3µm/s

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0

100000

200000

300000

400000

500000

600000

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Number of Counts

Position [µ m]

Position Histogram for φw≈55%

35mW22mW10mW

Figure 6. Histograms showing the distributions of particle position just below the glass transition

Figure 6 shows the histogram of particle position over time, in a sub-glassy (φ ≈ 55%) sample, which behavesas a dense but ergodic fluid. The only consequence of lowering the laser power is an increase in the standarddeviation of particle position, due to a decrease in trap stiffness. Although the probe particle is surrounded bya cage of nearest neighbours, rearrangements are relatively frequenct (there is a low energy barrier) and as theirpresence is not obvious in the position histograms even at the lowest powers, the trap potential is assumed todominate at this volume fraction. When compared to figure 7, the difference in behaviour is clear. This secondgraph shows the results of a static measurement in a glassy sample, where the trapped particle is confined notonly in the optical potential, but by its cage of nearest neighbours. At higher laser powers the trapped particleis pinned to a particular location in the suspension, but as the laser power is lowered the probe experiences cagerearrangments, relocating by a few tens of nanometers and then remaining in that position for some time before‘hopping’ again. This type of behaviour has been previously been observed in confocal microscopy studies2 andis not a tendancy to ergodicity, but an example of the ageing dynamics of a colloidal glass.

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0

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400000

500000

600000

700000

800000

-150 -100 -50 0 50 100

Number of Counts

Displacement [nm]

Displacement and Cage Hopping

55mW41mW15mW

Figure 7. Cage hopping at low trapping strength in a colloidal glass

The data in figure 8 show recordings of position over time for several different volume fractions at the samelaser power and same drag speed. The particle is held in the glass for a period of around 20 minutes, thendragging commences (all traces were recorded using a 35mW trap strength and 4µm/s drag speed). Manyinteresting features are present in this figure; firstly, there is a clear increase in average displacment with volumefraction; this is related to the yield stress of the material. The average displacement recorded can be relatedto an average force needed to drag the particle (the mean displacement during dragging) using the optical trapstiffness obtained during the calibration. When divided by the cross-sectional area of the probe particle, a valuefor the yield stress is obtained. The data at φ ≈ 61% and φ ≈ 63% shows a yield stress of around 4Pa and 7Parespectively, which is in good agreement with the findings of macrorheological studies,14 taking into accountparticle size and volume fraction.

This is quite remarkable when one considers that the mechanical behaviour of a suspension is the same whetherit’s examined as a group of around 104 particles (microrheology), or as a group of 1012 particles (macrorheology).Another salient feature of this graph is the shape of the peaks, and how this changes as a function of volumefraction. At low volume fractions, the displacements from equilibrium occur at a speed below the drag speed; thissuggests that the suspensin exhibits fluid-like behaviour in that it has a zero yield stress. At volume fractionsabove the glass transition (see the φ ≈ 61% and φ ≈ 63% traces in fig.8), a definite yield stress emerges,characterised by steep slopes, with a gradient corresponding to the the stage speed. This may be interpreted asa temporary structure forming that doesn’t allow the particle to pass, but is swept away when a higher forceis applied. There appears to be a ‘family’ of such features, of varying size and duration; it is hoped that awavelet-based analysis of the data will help justify this assertion.

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-100

0

100

200

300

400

500

600

700

800

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0 20 40 60 80 100 120 140

Displacement [

µm]

Time [s]

Displacement From Equilibrium as a Function of Time

φw≈55%φw≈57%φw≈61%φw≈63%

Figure 8. Particle displacement from equilibrium trap position as a function of time while being dragged in a densecolloidal suspension

4. CONCLUSIONS

Optical tweezers have been used to investigate the mechanical properties of colloidal suspensions from a micro-scopic perspective. Results obtained at low-volume fractions have been qualitatively explained and compared tosimilar results in a bare solvent medium. Results at higher volume fractions have shown the emergence of yieldstress and elastic behaviour in the suspension, which is consistent with the macrorheological data obtained byothers, notably the value of 4Pa for the yield stress of a φ ≈ 61% sample, and 7Pa for the φ ≈ 61% sample.

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8. A. Yethiraj and A. van Blaaderen, “A colloidal model system with and interaction tunable from hard sphereto soft and dipolar,” Nature 421, p. 513, 2003.

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