Colloids and Surfaces

A: Physicochemical and Engineering Aspects 190 (2001) 81–88

Position correlation microscopy: probing single particledynamics in colloidal suspensions

Stuart Henderson, Steven Mitchell, Paul Bartlett *School of Chemistry, Uni�ersity of Bristol, Cantock’s Close, Bristol BS8 1TS, UK

Abstract

A novel optical tweezer based microscope is described which probes the dynamics of an individual pair of colloidalparticles over many decades of time, with nanometer position resolution. The potential of the technique isdemonstrated by probing the distance dependence of the hydrodynamic forces between two polymer-coatedpoly(methyl methacrylate) spheres suspended in cyclohexane. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Optical tweezers; Hydrodynamics; Hard-sphere colloids; Brownian motion; Microscopy

www.elsevier.com/locate/colsurfa

1. Introduction

As our ability to control the bulk properties ofcolloidal systems has improved there has been arapid growth of interest in understanding themicroscopic processes which occur at the singleparticle level within a suspension. Traditional ex-perimental methods such as light, X-ray and neu-tron scattering generate average values fromessentially a very large number of copies of thesuspension. Such ensemble measurements inte-grate the information on the microscopic proper-ties of interest over the whole of thermal phasespace. Of course, the results are still of great valuebut they can only provide indirect insights intothe fundamental microscopic process which mostinterest us. Direct access to the forces and dynam-

ics of individual colloidal particles would allowfor a scrutiny of the microscopic processes at highresolution without ensemble averaging.

The task of manipulating and measuring theproperties of a single colloidal particle within asuspension is daunting. Electrostatic or entropiccolloidal forces on an individual particle are typi-cally of the order of 100 fN while the forces maychange substantially on a length scale of a fewnanometers. Any attempt to measure single col-loid properties requires therefore an instrumentwith high temporal and force resolution whichwill function in the presence of significant thermalmotion. Various methods have been developed inthe past for this task including optical [1,2], mag-netic traps [3] and atomic force microscopy [4]. Inthis paper a novel experimental technique is de-scribed which probes directly the microscopicforces acting on a pair of colloidal spheres overmany decades in time, with a sensitivity of a fewtens of femtonewtons and a sub-nanometer posi-

* Corresponding author. Tel.: +44-1225-826118; fax: +44-1225-826231.

E-mail address: p.bartlett@bristol.ac.uk (P. Bartlett).

0927-7757/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.

PII: S0927 -7757 (01 )00667 -7

S. Henderson et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 190 (2001) 81–8882

tion resolution. The position correlation micro-scope uses a pair of strongly focussed laser beamsto trap two colloidal spheres at a defined separa-tion. Thermal fluctuations in the positions of thespheres are measured by imaging the focused laserbeams onto a pair of quadrant photodetectors.Forces that act between the two spheres are deter-mined with high resolution by computing thecross correlation between the instantaneous posi-tion of each particle. Below we present the physi-cal implementation of this technique and describea simple experiment which demonstrates that thetechnique works.

2. Experimental methods

Fig. 1 depicts a schematic diagram of the exper-imental setup. The position correlation micro-scope is based around a standard invertedmicroscope (Axiovert S100, Zeiss) equipped withhigh numerical aperture optics. A pair of indepen-dent optical traps were formed by focussing twoorthogonally polarised beams from a Nd:YAGlaser (7910-Y4-106, Spectra Physics) with a wave-

length of �=1064 nm to diffraction-limited spotsin the sample using an oil-immersion microscopeobjective (100×/1.3 NA Plan Neofluar, Zeiss).The back aperture of the objective lens was overfi-lled to ensure high trapping efficiency. The powerof each laser beam, in the focal plane, was contin-uously adjustable up to a maximum of about 80mW. Laser intensities above 2 mW were justsufficient to hold the colloidal particles, used inthe current work, within the focus of the beam.The location of each polarised beam and thus thetrapped particle [5] was simply controlled by ad-justing the associated gimbal mounted mirror(GM) and the 1:1 telescope, formed by the pair oflens L1 and L2 detailed in Fig. 1. Altering thepositions of these components allowed the meanseparation of the two trapped spheres to be variedcontinuously between 2 and 30 �m. The exactpositions of the centres of the two traps weredetermined to within 40 nm using video mi-croscopy and a centroid tracking algorithm.

While the mean position of each particle wasfixed by the position of the corresponding laserfocus, fluctuating thermal forces caused small butcontinuous displacements of the particle away

Fig. 1. Schematic diagram of the dual-beam optical tweezer setup. The lens L1s and L2s form a 1:1 telescope for the S-polarised beamwhich together with the gimbal mounted mirror GMs are used to adjust the position of the S-polarised beam within the focal plane.The components L1p, L1p and GMp do the same for the P-polarised beam. P1, P2 and P3 are polarising beam splitting cubes, D1 andD2 are dichroic mirrors, Fs and Fp are 1064 nm line filters and QDs and QDp are the two quadrant detectors.

S. Henderson et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 190 (2001) 81–88 83

from the centre of the trap. For small displace-ments in the plane of the sample the opticaltrapping potential is approximately harmonic.The restoring force on the particle is then simplyproportional to the lateral displacement, F= −kL(x+y), with a lateral force constant kL whichfor a given particle and beam profile is a linearfunction of the laser power. In the experimentsdetailed below the trap spring constants werenormally of the order of 10−5 N m−1.

Measurements were performed on poly(12-hy-droxy stearic acid) stabilised poly(methylmethacrylate) (PMMA) spheres [6] of radius0.71�0.02 �m dispersed in a layer of cyclohexane170 �m thick. The volume fraction of particles inthe suspension was very low, of the order of5×10−6 to minimise the interference from strayspheres diffusing into the traps during measure-ments. A small concentration of free stabiliser wasadded to the suspension to minimise the adsorp-tion of colloidal particles onto the glass walls ofthe cell. The suspensions were contained inrectangular glass capillaries with a wall thicknessof 170 �m which were hermetically sealed with anepoxy resin to prevent evaporation.

On the microseconds timescale characteristic ofour experiments, the motion of a colloidal particleis heavily damped by viscous coupling with thefluid. Inertial terms are negligible and the netforce on a particle is essentially zero. Any col-loidal force on a particle is balanced by an equaland opposite optical restoring force. Conse-quently the thermal forces on a trapped colloidalparticle may be computed from the instantaneousposition of the particle. To determine the posi-tions of each of the two trapped spheres with highresolution the interference pattern between thetransmitted and scattered laser light was observedin the back-focal plane of the microscope con-denser [7]. Motion of a refractive particle in thehighly inhomogeneous field at the laser focussignificantly changes the amplitude of the forwardscattered light. Constructive interference with thetransmitted beam occurs in the back-focal planeof the condenser and causes intensity shifts whichare imaged by the lens L5 onto a quadrant detec-tor. The interference signals arising from the twoorthogonally polarised beams are separated using

a polarising beam-splitter (P3) and amplified bytwo custom-built current-to-voltage converters.Difference signals from the sum of the horizontal(X) and vertical (Y) halves of the quadrant detec-tors and the total signal from each photodiode arecalculated by analogue electronics before beingdigitised at a frequency of 20 kHz and stored forlater analysis. The position detectors were cali-brated by analysing the thermal fluctuations of atrapped Brownian particle. The high sensitivity ofthe quadrant diode detector is illustrated in Fig. 2,which shows the X-voltage response generated byscanning an immobilised particle in 10 nm square-wave steps, using a piezoelectric translation stage(P-517 2CL, Physik Instrumente), through thetrap centre. As is clear, the noise on the detectorsignal is at the nanometer level which for a typicaltrap stiffness of kL�10−5 N m−1 equates to anequivalent force sensitivity of �10 fN.

3. Single particle experiments

To calibrate the force measurements the ther-mally driven position fluctuations of a single iso-lated particle were studied. Probably the simplestmeasure of the dynamics of a trapped Brownianparticle is the mean-squared displacement a parti-cle undergoes in a time interval �

��x2(�)�=�[x(t+�)−x(t)]2� (1)

where x(t) is the instantaneous position of theparticle after a time t and �···� denotes a time-av-eraged quantity. The mean-squared displacement��x2(�)� as a function of � is readily calculatedfrom the measured displacement record x(t). It isobvious that for �=0 the mean-squared displace-ment must start at ��x2(�)�=0 while in a fo-cussed beam, since the particle is confined to arestricted region of space, the mean-squared dis-placement will at long times approach a plateauvalue ��x2(���)�. Fig. 3 shows the time-depen-dence of ��x2(�)� measured in a 52 s period atthree different laser intensities for a 0.71 �mparticle trapped in a S-polarised beam 20 �mabove the wall of the capillary. It is clear that themeasured data shows the time-dependence ex-pected with the plateau value ��x2(���)� in-

S. Henderson et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 190 (2001) 81–8884

Fig. 2. The top trace shows the X-detector response in millivolts as an immobilised 0.71 �m radius PMMA particle mounted on apiezoelectric stage is driven along the x-axis through the centre of the focussed beam with a�5 nm amplitude square wave. Thelower trace shows the driving voltage.

creasing with decreasing laser intensity. Interest-ingly, it is also clear that at short times ��x2(�)�decays linearly with a slope that is independent oflaser intensity. To quantitatively interpret theseobservations we assume that the trapping poten-tial is harmonic. The dynamics of a Brownianparticle bound in a harmonic potential have beenanalysed in detail by Uhlenbeck and Ornstein [8].The problem has two characteristic time scales, �f

and �s, which define the decay time � f−1 of the

velocity autocorrelation function of the Brownianparticle and the time �s on which the displacementof the Brownian particle is influenced strongly bythe potential

�f=m/6��a (2)

�s=6��a/kL

Here m is the effective mass of the colloidalparticle, � the viscosity, a the particle radius, andkL the force constant of the harmonic potential. Itis instructive to estimate the relevant time scales

for the experiments detailed here. For a 0.71 �mradius PMMA particle in cyclohexane, �f is about2×10−7 s and all of the current measurementsare therefore made in the ‘long-time’ limit ���f

where inertial terms are negligible. Secondly, forthe harmonic trap stiffnesses used in the currentwork (typically of order 10−5 N m−1) �s is of theorder of 10−3 s. Consequently the fast decay time�s is much shorter than the slow decay time �s andthe dynamics of the Brownian particle arestrongly overdamped [9]. In this limit, it isstraightforward to demonstrate that for ���f themean-squared displacement is given by theexpression

��x2(�)�=��x2(���)�

�

1−exp�

−2D0�

��x2(���)��n

(3)

with the free diffusion coefficient, D0=kBT/6��aExpansion of Eq. (3) in powers of � shows that atshort times, where �f����s, the particle diffuses

S. Henderson et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 190 (2001) 81–88 85

freely and the mean-squared displacement growsinitially linearly with time, ��x2(�)�=2D0�. Overlonger times the diffusive motion of the particle isgradually retarded by the harmonic potential andthe mean-squared displacement eventually satu-rates at the equilibrium limit ��x2(���)�=2kBT/kL. The solid lines in Fig. 3 show the resultsof a numerical fit of Eq. (3) to the experimentaldata with a fixed diffusion constant D0 calculatedfrom the particle radius and the viscosity at 25°Cof cyclohexane (�=0.894×10−3 Pa s). It is clearthat the harmonic model accounts accurately forthe experimental data over some two decades overtime and at three different laser intensities.

4. Two particle experiments

A colloidal particle held within an optical trapis subject to continuous, small but measurablefluctuations in position as a result of Brownianforces. As the trapped particle moves a flow iscreated in the surrounding fluid which causesneighbouring particles to move [10]. Such hydro-dynamic coupling is of tremendous importance in

Fig. 4. The geometry of the dual-trap experiments. A pair ofspheres is held in the orthgonally polarised S and P traps,spaced a distance r along the x-axis. The arrows show thedirection of the optical gradient forces, F= −kLx, when asphere is displaced a distance x from the centre of the trap.

determining the aggregation, rheology and sedi-mentation of colloidal suspensions [11]. Yet de-spite the prevalence and significance of suchforces our current theoretical and experimentalunderstanding of hydrodynamic effects in all butthe simplest colloidal systems is either sketchy orat best controversial [12]. Theoretical and numeri-cal difficulties stem from the long-range nature ofhydrodynamic forces while until recently [1] ex-perimental insights have been limited becauseconventional techniques such as dynamic lightscattering provide only highly averaged ratherindirect information on hydrodynamic interac-tions. Here we describe how optical trapping tech-niques may be used to measure directly thepair-mobility tensor for two polymer-coated col-loidal spheres as a function of the centre-to-centreseparation r.

Fig. 4 shows schematically the geometry of ourmeasurements. A pair of polymer-coated PMMAspheres of radius a=0.71 �m, dispersed in cyclo-hexane, were held in two optical tweezers at anear constant separation r, measured along thex-axis and spaced at least 20 �m from any wall.The separation between the two traps was ad-justed between 2.5 and 20 �m. The power in eachof the orthogonally polarised beams was carefullyadjusted so that the stiffness of the two opticaltraps were identical. Each beam had a power oftypically 30 mW which resulted in a trap stiffnessof 9.1×10−6 N m−1. Individual spheres moveonly with a rms amplitude of 21 nm in the trap so

Fig. 3. The time dependence of the mean-squared displacement��x2(�)� of a 0.71 �m PMMA sphere trapped in beams of 13mW (squares), 18 mW (filled circles), and 43 mW (opencircles). 220 data points were collected at a sampling interval of50 �s. The solid lines are numerical fits using the expression ofEq. (3) and a free diffusion constant D0=3.46×10−13

m3 s−1. The mean-squared displacement of an untrappedsphere is shown dashed.

S. Henderson et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 190 (2001) 81–8886

to a good approximation the separation betweentwo trapped spheres is constant and equal to thetrap separation r. The location of each sphere,x1(t) and x2(t), in the focal plane at the time t wasmeasured with nanometer precision using the cali-brated quadrant photodetector system describedin Section 2. Similar colloidal systems have beenstudied by a number of authors [13,14]. Theseprevious studies have established that the inter-particle potential is steep, purely repulsive, andwell approximated by a hard sphere interaction.Consequently when the two traps are spaced by atleast one colloidal diameter (r�2a) no potentialinteraction between the two spheres is expected.Under such conditions the motions of the twotrapped spheres are not, however, independent.Thermal fluctuations in the position of eachsphere causes the suspending fluid to flow whichwill influence the position of the second particle.The degree of coupling between the position fluc-tuations of the two spheres depends on the magni-tude of the hydrodynamic forces measured at thefixed trap separation r. Below we detail how thehydrodynamic drag forces may be investigatedfrom a measurement of their effect on the cross-correlation between the position fluctuations ofthe two trapped spheres.

The dynamics of the two trapped spheres aremost readily understood in terms of the nor-malised cross correlation function h12(�) which isdefined by the equation

h12(�)=(x1(t+�)x2(t))

��x1(t)2��x2(t)2�(4)

where x1(t) and x2(t) are the measured x-posi-tions of sphere one and two, respectively, at atime t. h12 summarises the statistically averagedcorrelations between the positions of the twospheres. If, for instance, the motions of the twoparticles are independent of each other then h12=0 while a positive value signifies that the positionfluctuations of the two spheres are, on average,in-phase with a negative value indicating con-versely that the position fluctuations are anti-cor-related. The asymptotic dependence of h12(�) onthe delay time � can be derived from a fewstraightforward considerations. At long delaytimes, ���, the position of the two spheres will

be uncorrelated and h12(���) will equal zero asthe hydrodynamic flows which couples the motionof the two spheres together are dissipative and somust decay to zero at long times. Similarly atshort times, ��0, the cross-correlation is alsozero as h12(�=0) records only static correlations[9] between the particles which, at thermal equi-librium, are a function only of the interparticlepotential. In the current experiments there is nopotential coupling between the two spheres (at alltrap separation r�2a) and so h12(�=0)=0. Thecross correlation function is therefore zero atshort and long times and non-zero only for inter-mediate times.

Fig. 5 shows experimental measurements of thecross-correlation functions at three different trapseparations r. For a given separation the trappedspheres were followed for a total of 420 s yieldingin excess of 8×106 samples of their dynamics inintervals of 50 �s. The displacement record wasdivided into sequences 0.8 s long and the crosscorrelation function h12(�) was computed by aseries of fast Fourier transforms. Two features inthe experimental data are striking. First, the datashows, rather surprisingly, that hydrodynamic in-teractions cause the two particles to be anti-corre-lated at intermediate times and second, the degreeby which the two particles are anti-correlatedincreases markedly as the separation between thetwo particles reduces.

Fig. 5. The normalised cross-correlation h12 as a function ofdelay time �, measured at three centre-to-centre separations �.Squares, r=15.37 �m; circles, r=4.19 �m; triangles, r=2.50�m. Every eighth data point is plotted.

S. Henderson et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 190 (2001) 81–88 87

The hydrodynamic forces acting between twospheres in low-Reynolds-number flow have beencalculated exactly by a number of authors [15,16].A key quantity is the mobility matrix �ij whichspecifies the instantaneous velocity of particle iwhen an external force Fj is applied to particle j,through the linear expression x� i=�ij Fj. For asingle isolated sphere of radius a in a fluid ofviscosity � the mobility coefficient is, from theStokes equation [10], simply �=1/6��a. When asecond sphere is added, the situation becomessignificantly more complicated. The sphericalsymmetry of the Stokes limit is reduced to axialsymmetry and the mobility matrix depends cru-cially on the geometry of the two spheres [15].Limiting the present discussion to motion alongthe line of centres, then it is clear that the off-di-agonal element �12 couples the motion of the twospheres together and the coordinates x1 and x2 areno longer independent. The motion of the coupledspheres can be decomposed into two independentmodes by introducing the normal coordinates

Xs=1

�2(x1+x2) (5)

Xa=1

�2(x1−x2)

which describe in turn a symmetric (Xs) commonmotion of the centre of mass of the two spheresand an anti-symmetric (Xa) relative motion of thetwo spheres with respect to each other. The twocoordinates Xs and Xa are independent of eachother. The corresponding symmetric (�s) and anti-symmetric (�a) mobilities can be calculated fromthe asymptotic expressions, valid at large separa-tions, given by Batchelor [15]

�s=1

6��a�

1+3

2�−

1�3−

154�4+O(�−6)

�(6)

�a=1

6��a�

1−3

2�+

1�3−

154�4+O(�−6)

�with � the dimensionless centre-to-centre separa-tion, �=r/a. Examination of the lowest orderterms shows that the mobility of the symmetricmode is enhanced and the anti-symmetric modereduced in comparison with an isolated particle.The reduction in mobility of the anti-symmetric

mode reflects the increased resistance of the fluidto being squeezed out from the narrow gap be-tween two approaching spheres while the in-creased mobility for the symmetric mode is causedby the tendency for the fluid flow generated byone sphere to drag along a neighbouring sphere.

The asymmetry in the mobilities of the twonormal modes seen in Eq. (6) is reflected in thepronounced anti-correlation observed in the ex-perimental data. The origin of the observed corre-lation may be understood from a simple physicalargument. Thermal fluctuations in the positions ofthe two spheres may be decomposed into indepen-dent fluctuations in the two normal modes. Sym-metric fluctuations contribute positively to thecross correlation function h12 while the anti-sym-metric fluctuations make a negative contribution.At �=0 the cross correlation function h12 is zero(because there is no potential acting between thetwo spheres) so the proportion of symmetric andanti-symmetric fluctuations must be equal at �=0. In the case where the two spheres are widelyseparated Eq. (6) demonstrates that the mobilitiesand hence the decay times for the two modes areidentical. Consequently while the contribution ofeach of the two modes to the cross correlationdecays with time their sum remains at zero andh12(�)=0 for all times �. However, when the twospheres are in close proximity to one another themobility of the symmetric mode is enhanced com-pared with the anti-symmetric mode. Correspond-ingly, the decay time for symmetric fluctuations isshorter than the decay time for anti-symmetricfluctuations. As a consequence the cross correla-tion function develops a pronounced anti-correla-tion at intermediate times due to the presence oflong-lived anti-symmetric fluctuations.

In conclusion, the present study demonstratesthat the hydrodynamic coupling between an indi-vidual pair of hard-sphere colloids can be accu-rately measured using a dual-beam opticaltweezer. The experimental data shows, rather sur-prisingly, that the motions of the two particles areanti-correlated at intermediate times. This obser-vation may be understood in terms of the asym-metry of the normal modes for a pair of particles.

S. Henderson et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 190 (2001) 81–8888

Acknowledgements

This work was supported by the Engineeringand Physical Sciences Research Council throughgrant GR/L37533.

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