Rheol ActaDOI 10.1007/s00397-013-0698-2

ORIGINAL CONTRIBUTION

Calibration of an optical tweezer microrheometerby sequential impulse response

Matthew M. Shindel · James W. Swan · Eric M. Furst

Received: 7 September 2012 / Revised: 21 January 2013 / Accepted: 28 February 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract We report a robust method for calibrating opticaltweezers in any viscoelastic medium. This approach usestwo coupled measurements—one from a static experimentin which a trapped particle diffuses passively within thetweezer’s harmonic potential and another from a dynamicexperiment in which the trap is jumped discontinuouslyto a new position while the particle undergoes transientrelaxation back into the minimum of the optical potential.Together, these are sufficient to determine the stiffness ofthe trap in a material of unknown rheology. The method istested in a Newtonian fluid and compares favorably withother means of calibration. The calibration is also per-formed in a non-Newtonian fluid of which standard opticaltweezer calibration methods may struggle to characterize.The correctly calibrated optical tweezer microrheometermeasures the rheology of polymer solutions in agreementwith macrorheological measurements.

Keywords Optical tweezers · Microrheology ·Viscoelastic media

Introduction

Complex fluid rheology underpins numerous physical andbiological processes—the result of an intimate associa-tion between the microstructure and the stress response of

Special issue devoted to novel trends in rheology.

M. M. Shindel · J. W. Swan · E. M. Furst (�)Department of Chemical and Biomolecular Engineering,Center for Molecular and Engineering Thermodynamics,University of Delaware, 150 Academy St.,Newark, DE 19716, USAe-mail: furst@udel.edu

soft materials. The dynamics and response to deformationof soft matter dictate both mechanical stability and pro-cess ability. Consequently, rheological characterization isimperative in the design of advanced manufacturing pro-cesses, particularly microfabrication, and in the engineeringof nano-structured, functional materials.

Microscopic rheometry has increased in prevalence asa class of techniques for determining the viscoelasticityof materials. The minimal sample volume required andextended range of frequencies probed are marked advan-tages. One approach, passive microrheology, extracts thematerial’s linear viscoelasticity, the complex shear modu-lus, from thermal fluctuations of embedded colloidal tracers(MacKintosh and Schmidt 1999; Squires and Mason 2010).Acquisition of statistically relevant information at a givenfrequency requires the magnitude of the ensemble-averagedprobe fluctuation to be greater than the measurement’snoise limit. In sufficiently stiff materials, Brownian pertur-bations alone cannot satisfy this criterion. In such cases,colloidal probes can be actively driven in order to discernthe linear viscoelastic response. This is active microrhe-ology, which enables investigation of both the linear andnonlinear response of a material depending on the magni-tude of the external driving force (Furst 2005; Brau et al.2007; Mizuno et al. 2008). Single-particle active microrhe-ology can be performed with the localizing photonic forcegenerated by optical tweezers. However, the stiffness of theoptical trap, and often the particle tracking system, mustbe calibrated prior to every experiment. This is particularlydifficult in materials with unknown rheology.

Optical tweezers are also often employed for measure-ment of pico-Newton scale forces in soft materials. Thismay encompass measuring thermal and hydrodynamic col-loidal pair interactions or tracking the dynamic response ofmacromolecules under mechanical stress (Park and Furst

Rheol Acta

2011; Lele et al. 2011; Neuman and Block 2004) and canbe performed in both Newtonian and non-Newtonian media.The quality of such measurements is, again, dictated bythe accuracy of the calibration. Such an approach mustwork quickly in temporally evolving environments. That is,effects such as irreversible aggregation, sedimentation, andevaporation, among others, can limit the amount of time onehas to complete an experiment. The sample must remainviable for much longer than it takes to perform the calibra-tion, as the experiment of interest is the quantification ofinteraction potentials, not the rheological characterization ofthe medium itself. Such temporal constraints apply equallyto microrheology.

The sequential impulse response (SIR) method reportedhere is designed to enable such flexible, rapid, and tractablecalibration. It does not require advanced knowledge of themedium’s viscoelastic properties or the trapping charac-teristics of the tweezing system and is, therefore, quitegeneral. Two simple and expeditious experiments—one inwhich passive diffusion of a trapped particle is observedand another in which the tweezers are moved discontin-uously and the relaxation of the particle into the trap isrecorded—are sufficient for determination of the trap stiff-ness in a medium with an unknown viscoelastic spectrum.The article is organized as follows. In Section “Equipmentand materials”, we describe the optical tweezers and themedia used in validating the SIR technique. In Section“Methodology”, the physical details underlying SIR areexplained. Section “Results” describes the results of SIRcalibration in water and draws comparisons with otherapproaches to tweezer calibration. Calibration results in anon-Newtonian fluid are also presented. Finally, in Section“Conclusions”, we provide a brief discussion of relevant

measurement limitations associated with optical tweezersand delineate operational boundaries which define a regimewhere the SIR method is most accurate.

Equipment and materials

Optical tweezers

A diagram of the optical tweezers is shown in Fig. 1. Trap-ping radiation emanates from a 4-W CW neodymium-dopedyttrium aluminum garnet (Nd:YAG) near-infrared (NIR,λ = 1,064 nm) laser (Coherent Compass, 1064-400m), andis first directed through an optical train that polarizes, col-limates, and expands the initial beam by a factor of 5. Thelaser then passes through a pair of orthogonal acousto-opticdeflectors (AOD; AA Optoelectronics, AA.DTS.XY-400),used to manipulate trap position in the sample plane, fol-lowed by a ∼1:1 telescopic lens pair used to re-collimatethe beam and conjugate the AOD with the objective back-focal plane. A periscope, composed of dielectric mirrorsand situated in the middle of the final telescope, is used toelevate the beam path, circumventing the internal optics ofan inverted optical microscope (Zeiss, Axiovert 200). Themicroscope is retrofitted with an IR-reflective dichroic mir-ror below the objective turret. This redirects the laser intothe back aperture of a 63×, 1.2 N.A., water immersionobjective (Zeiss, C-Apochromat). The objective creates anoptical trap within the sample by focusing the laser to adiffraction-limited spot.

We employ back-focal plane interferometry to track themotion of objects confined within the optical trap (Gittesand Schmidt 1998). A long working distance, 40×, 0.6 N.A.

Fig. 1 Optical tweezerschematic. Dashed outlinerepresents the microscope

Rheol Acta

objective serves as the condenser, collecting and recolli-mating the laser light after it passes through a sample. Asecond dichroic mirror reflects the collected light onto arelay lens positioned between the condensing unit and aquadrant photodiode (QPD; First Sensor, QP45-Q-HVSD).The relative spacing between the condenser’s back-focalplane, relay lens and QPD satisfies the thin lens equationwith 75 % demagnification. The QPD thus resides in a planeconjugate to the condenser’s back-focal plane. The inten-sity and spectral purity of light reaching the QPD is tunedby an adjustable pair of neutral density filters and a nar-row bandpass filter (1,064 ± 2 nm). The condensing anddetection optics are mounted on the microscope’s translat-able illumination pillar. This branch can be repositionedfor either interferometric tracking or imaging purposes.The former is engaged by conjugating the objective andcondenser back-focal planes while the latter is optimizedthrough Koehler illumination, achieved by conjugating thecondenser back-focal plane with the imaging plane.

The QPD is composed of Q-type silicon for enhancedquantum efficiency in the NIR portion of the electromag-netic spectrum. This eradicates lag due to the sluggishevolution of charge carriers typically inherent to the com-bination of NIR lasers and silicon detectors (Berg-Sørensenet al. 2003). The sensor is subjected to a forward operatingbias of 48 V, enabling a maximum bandwidth of 150 kHz.The photodiode is mounted on a printed circuit board con-taining four trans-impedance amplifiers that convert thephotocurrents generated in each of the sensor’s segmentsto voltages. A pair of differential amplifiers, also installedon the board, combine the individual quadrant signals intodifferential voltages, the values of which are in proportionto the relative displacement between the particle and trapalong the in-plane (x and y) and out-of-plane (z) directions.The latter is proportional to the sum signal from all fourquadrants. Prior to acquisition, an analog divider circuit nor-malizes the x and y voltages from the QPD module by thesum signal and rescales the position values by a factor of10 V. This reduces shot noise and ensures that bounds ofthe signal’s magnitude are compatible with our acquisitionhardware.

A white light-emitting diode above the second dichroicmirror illuminates the sample for bright-field imaging. Thefield of view magnified by the immersion objective is dig-itally rendered by a monochrome CCD camera (Hitachi,KP-M1AN, 30 fps) connected to the front port of themicroscope.

The voltage signal streamed to the AOD controller andthose emanating from the QPD are generated and acquired,respectively, by a data acquisition card (DAQ; NationalInstruments, NI PCI-6221) installed in a PC via an internalPCI interface bus. The position of the optical trap is linearlydependent on the voltage applied to the AOD controller.

Proportionality constants of 11.86 and 11.31 μm/V weremeasured for the two orthogonal channels prior to exper-iments. Once generated, the AOD signals are re-digitizedand acquired through the card’s internal channels. The cardcontains a single analog-to-digital converter which is sharedby all active analog inputs, effectively synchronizing therecorded AOD and QPD voltage signals.

The optical tweezers are operated with control softwarewritten in the LabView graphical programming environ-ment (National Instruments). The software consists of twoparallel loops, one dedicated to signal generation and theother to acquisition and data streaming. The generationloop creates a periodic voltage waveform that is exportedto the AOD controller. The acquisition loop is responsi-ble for storing AOD and QPD signals in dynamic memoryand then streaming the acquired data to the computer’s harddisk. The data presented here were recorded at a bandwidthof 40 kHz.

Sample preparation

Sample cells were constructed by affixing a coverslip (no.1.5) to a microscope slide with two pieces of double-sidedtape. We use water as our model Newtonian fluid. Cellswere loaded with an aqueous dispersion of polymethyl-methacrylate particles (average diameter = 1.57μm; BangLaboratories) at a volume fraction of φ ∼ 10−4, and thensealed with optical adhesive. The solution is drawn intothe cell via capillary action. Aqueous poly(ethylene oxide)(PEO; Sigma-Aldrich) with a molecular weight of 200 kDawas used as a model viscoelastic medium. PEO solutionsat three concentrations (1.25, 2.5, and 5% w/w) were pre-pared by dissolution in water and subsequent dilution. Themixtures were stored at 50 ◦C for 10 days to dissolve thepolymer. Sodium azide (0.02% w/v) was added to inhibitbacterial growth. Residual, insoluble elements (primarilybutylated hydroxytoluene) were removed by centrifugation.Once equilibrated, a trace amount of polymethylmethacry-late (PMMA) particles was added to the solutions (volumefraction φ < 10−4).

Methodology

The SIR calibration technique is performed in two stagesonce a particle is trapped, as depicted in Fig. 2. Initially,the trap position, denoted X(t), is held constant while theQPD captures the equilibrium fluctuations of the trappedparticle’s position, denoted w(t). This stage is termed static.The spectrum of the particle’s fluctuations constitutes onecharacterization of the material’s linear viscoelasticity andthe stiffness of the laser trap. During the second stage,the laser is repeatedly “jumped” along a single axis by

Rheol Acta

Fig. 2 Position of the optical trap as a function of time (top) and cor-responding particle displacement (bottom). The units of the raw timeseries data are indicated. The inset shows the circled peak in the detec-tor signal (black) overlaid on the corresponding portion of the trap’spositional time series (red). Both curves have been normalized, so theyreside on the same scale. The QPD displays a local maximum at theinstant a step change is applied to the trapping voltage. The two datastreams are recorded synchronously so long as the sampling rate isbelow the QPD’s maximum update rate (150 kHz)

applying an amplitude-modulated square wave to one ofthe AOD channels. Here, each discrete translation in trapposition can be described mathematically by the Heavi-side step function, H(t). This stage is termed dynamic. Forsuitably small jumps, the response of the particle repre-sents a second, independent characterization of the linearviscoelasticity and trap stiffness. Additionally, with highenough sampling frequency, the sensitivity of the QPD maybe extracted from the instantaneous voltage values acquiredimmediately after the jump occurs (Le Gall et al. 2010). Weshall demonstrate that the power spectrum of fluctuations,when used in conjunction with the transient response (TR)to jumping, is sufficient to determine simultaneously the

medium viscoelasticity and the stiffness of the optical trap.Fischer et al. previously proposed a similar tweezer calibra-tion scheme comprising passive and active measurements(Fischer et al. 2010). However, their procedure involves cal-ibrating the detector by rastering the laser over a particleimmobilized on the sample cell wall. Consequently, detec-tor sensitivity and trap stiffness are not determined with thesame probe nor at the same vertical location in the sample.

A software-based trigger is used to gate the waveformapplied to the AOD, enabling seamless toggling betweenthe static and dynamic modes (see Fig. 2). The static periodof the calibration is typically made to last for 1–2 min inlow-viscosity fluids such as water. The jump frequency indynamic mode was set at 20 Hz (yielding a square wave fre-quency of 10 Hz) for our water-based experiments, and 50different jump distances were each repeated 50 times. Thus,the entire calibration procedure required ∼ 3–4 min. Thistime period can be reduced by operating at a higher jumpfrequency. The full TR must be sampled to ensure accuratecalibration. This requires that the jump frequency be muchlower than the trap’s corner frequency, given by the ratio oftrap stiffness to the Stokes drag coefficient, 6πηa, where a

is the particle radius and η is the zero shear viscosity of themedium. This relation allows the particle to relax back intothe trap before another jump begins. Although stiffness, vis-cosity, and particle size may not be known a priori, an upperbound for the jump frequency can be obtained with order-of-magnitude estimates for these parameters. The opticaltrap stiffness for our tweezing system and the viscosity ofaqueous media typically reside on the order of 10 pN/μmand 1 cP, respectively. Therefore, the trap’s corner frequencyis on the order of 102 Hz when holding a 1μm particle.Accordingly, the jump frequency should be set an order ofmagnitude lower than the corner frequency providing theparticle with a sufficient amount of time to relax back to anequilibrium state.

The AOD settling time is ∼ 1μs. Once the trap isjumped, the particle relaxes back into the minimum ofthe optical potential on a time scale proportional to theinverse corner frequency (∼ 10 ms in aqueous media andslower with increasing viscosity). The disparity between therespective time scales for trap motion and particle relax-ation produces delta-like peaks in the QPD voltage signal,the maxima of which correspond to the prescribed jump dis-tance. Brownian motion slightly offsets the particle’s initialdisplacement for each jump, but this deviation can be aver-aged out through repetition of the jumping procedure. Theensemble average of the dynamic experiment is insensitiveto thermal fluctuations. The detector sensitivity is deter-mined from a linear fit of the mean peak height as a functionof jump distance.

Rheol Acta

Micromechanical model

In the linear response regime, the average force an isotropicand homogenous material exerts on a spherical colloidalparticle with center at x(t) is

FH = −∫ ∞

−∞R(t − t ′)x(t ′) dt ′. (1)

R(t) is termed the resistivity which describes the historydependence of the hydrodynamic force on motion of the col-loid. Neglecting inertia, the generalized Langevin equationgoverns the balance of forces on the particle

0 = −∫ ∞

−∞R(t − t ′)x(t ′) dt ′ − κ

(x(t) − X(t)

) + FB(t). (2)

where FB(t) is the stochastic Brownian force, and we havesupposed the particle is trapped in a harmonic potential withX(t), the position of the trap and κ , the trap stiffness. Theforce balance (Eq. 2) can be recast in terms of the particle’sdisplacement from the trap w(t) = x(t) − X(t)

0 = −∫ ∞

−∞R(t − t ′)

(w(t ′)+ X(t ′)

)dt ′ −κw(t)+FB(t). (3)

The Brownian force acts as a random, stationary pro-cess, and thus has zero mean and autocorrelation in time〈FB(t) ·FB(t ′)〉 = 2kBT R(|t − t ′|) given by the fluctuation-dissipation theorem (kB is Boltzmann’s constant and T

denotes absolute temperature).

Static measurement

During the static measurement, the trap is steady so thatX(t) = 0. The Fourier transformation

ˆ(·) =∫ ∞

−∞(·) exp(if t) dt, (4)

of Eq. 3 is

0 = −(if R(f ) + κ)w(f ) + FB(f ), (5)

and therefore, the power spectral density (PSD) of fluctua-tions in w(t) is

S(f ) = 2kBTR{R(f )}∣∣if R(f ) + κ∣∣2

. (6)

In a purely Newtonian medium, the resistivity is given bythe Stoke’s drag, R(t) = 6πηaδ(t), reducing S(f ) to aLorentzian distribution which can be fit directly in orderto extract the trap stiffness and solvent viscosity (Berg-Sørensen and Flyvbjerg 2004; Neuman and Block 2004).In a viscoelastic medium, an explicit representation of theresistivity is typically unknown. However, by stipulatingthat the material exhibits some viscous dissipation when

steadily deformed, we may conclude that the power spec-trum is finite and that f R(f ) = 0 as f → 0. Thisallows approximation of the PSD at low frequencies by asecond-order Taylor expansion

S(f ) ≈ S(0) + 1

2S′′(0)f 2, (7)

where

S(0) = 2kBT

κ

{R(0)

κ

}, (8)

and

S′′(0) = −2kBT

κ

{2

(R(0)

κ

)3

+ 4iR(0)

κ

R′(0)

κ− R′′(0)

κ

}.

(9)

The first-order derivative in the expansion of Eq. 7 must bezero as S(f ) is an even function of frequency. We shall showthat the bracketed terms in Eqs. 8 and 9 can be determinedfrom the TR of the particle following a laser jump.

Dynamic measurement

In modeling the jump experiment, we represent the positionof the trap as a step function X(t) = hH(t). Therefore, theFourier transformation of Eq. 3 is

0 = −R(f )(if w(f ) + h

) − κw(f ) + FB(f ). (10)

The Brownian force may be neglected when the response isaveraged over many realizations. Solving for R(f ), we findthat

R(f ) = − κg(f )

if g(f ) + 1, (11)

where g(f ) = (〈w(f )〉 ·h)/(h ·h) is the ensemble-averagedTR normalized by the impulse magnitude. In the limit thatf → 0, it follows from Eq. 11 and the definition of theFourier transformation that{

R(0)

κ

}= −g(0) = −

∫ ∞

0g(t) dt. (12)

The lower limit of integration is set by the fact that g(t) = 0for t < 0 by definition. Following the same line of analysisreveals

−{

2

(R(0)

κ

)3

+ 4iR(0)

κ

R′(0)

κ− R′′(0)

κ

}

= −g′′(0) =∫ ∞

0t2g(t) dt. (13)

These are precisely the bracketed terms in Eqs. 8 and 9, sothat the TR supplements the expansion of the PSD.

Rheol Acta

Determination of trap stiffness

Substitution for these bracketed terms in Eq. 7 relates theTR and the PSD:

S(f ) ≈ kBT

κ

[− 2

(∫ ∞

0g(t) dt

)+ f 2

(∫ ∞

0t2g(t) dt

) ].

(14)

In Eq. 14, there remains a single, unknown coefficient:kBT /κ . The trap stiffness is determined from a one-parameter fit of the PSD in the limit of low frequencies(i.e., below the corner frequency). Note that this approachdoes not require an explicit form for the resistivity nor priorknowledge of the the trap stiffness. The only assumptionsnecessary were based on linear response in both the vis-coelasticity of the medium and the trapping force. Withthe trap stiffness determined, the zero frequency limit ofthe resistivity may be found directly from the integratedTR, viz.

R(0) = −κ

∫ ∞

0g(t) dt. (15)

Higher order derivatives of the resistivity at zero frequencymay be extracted similarly; for example,

R′(0) = −iκ

[∫ ∞

0tg(t) dt −

(∫ ∞

0g(t) dt

)2]

. (16)

For a single-mode Maxwell fluid, R′(0) may be relateddirectly to the characteristic elastic modulus in much thesame way R(0) relates to the characteristic viscous modu-lus. Therefore, these may prove to be useful measures ofnon-Newtonian behavior.

The full frequency-dependent resistivity or equivalentlythe complex modulus, G(f ) = if R(f )/6πa, can beextracted from either the PSD or the Fourier transforma-tion of the TR through methods described by many, notably(Brau et al. 2007; Preece et al. 2011; Pesce et al. 2009). Theapproaches presented in these articles require independentdetermination of the trap stiffness. With the SIR method, thetrap stiffness can also be measured through careful analysisof these same response functions.

Results

Calibration in Newtonian fluids

Accuracy of the SIR calibration was tested in water. A sin-gle PMMA particle was trapped, and the calibration wasperformed at five different laser powers and at 24 ◦C. Thedetector response curves depicted in Fig. 3 have an S-shape,typical of position-sensitive photodiodes. Sensitivity, given

Fig. 3 Detector response curves obtained at five different laser powerswith a 0.69-μm radius PMMA particle immersed in water. The curveshave been vertically offset for visual clarity. Detector sensitivity wasextracted from a linear fit over a range of ± 300 nm in each case

by the slope of the linear portion of the response, was∼ 100 nm/V in all cases.

After calibrating the detector, QPD voltage signalsrecorded during the static and dynamic measurements arelinearly transformed into the relative particle displacement,w(t). The displacement from the dynamic segment of thecalibration is parsed into individual periods (each cyclecontaining a single jump in trap position) and the normal-ized TR, g(t), is taken from the average and subsequentlyintegrated in accordance with Eqs. 12 and 13. Only mea-surements with an impulse magnitude within the detector’slinear range (|h| ≤ 300 nm) are used. The linear range of thetrapping potential is known to be larger than 300 nm. ThePSD (see Fig. 4), calculated from the results of the staticexperiment, is used to determine trap stiffness by fitting forthe leading coefficient in Eq. 14, kBT /κ .

In a viscous fluid, the PSD has the form of a Lorentziandistribution given by replacing R(f ) in Eq. 6 with theStokes drag coefficient, 6πηa:

S(f ) = 2kBT

6πηa(f 2c + f 2)

(17)

where fc = κ/(6πηa) may be identified as the corner fre-quency. At high frequencies (f � fc), the power spectrumdecays as ∼ f −2, characteristic of short-time diffusion. ThePSDs at each power collapse in this limit, as the viscosity ofthe medium is invariant. In contrast, over long times (f �fc), the optical trap limits particle motion and the inten-sity of the spatial fluctuations asymptotically approaches aplateau value as f → 0. Fitting the PSD to a Lorentzian is acommon means of trap calibration in viscous media, and weemploy this method for comparison with the SIR calibrationresults (Berg-Sørensen and Flyvbjerg 2004). However, we

Rheol Acta

Fig. 4 Power spectra obtained at five different laser powers for aPMMA particle immersed in water. The measured spectra come fromwindowing over ∼ 7-s periods. The dashed lines correspond to theplateau values predicted by the SIR calibration. They indicate whereand to what extent low-frequency noise contaminates the spectrum.Solid lines are fits of Eq. 14 for f ∈ [fc/7, fc/4]

note that this method has limited applicability in viscoelas-tic fluids since the functional form of the time-dependentresistivity is typically unknown.

The analog signal processing circuits, power electron-ics, and AOD controller impart harmonic noise to the PSDat high frequencies. Aliasing caused by the finite sam-pling frequency also distorts this region of the PSD. Laserwobble, drift, and statistical artifacts can contaminate thelow-frequency regime of the spectrum (see Fig. 4), givingrise to ballistic noise which adopts the typical scaling off −1. This noise increases in relative strength with increas-ing laser power. Although the accuracy of the quadraticapproximation to the PSD is greatest near the point ofexpansion (f = 0), we take care to avoid extracting datafrom these very low-frequency regions by constraining thelower bound of the fitting range. Here, we fit Eq. 14 onlyfor f ∈ [fc/7, fc/4], as shown in Fig. 4. A precise valueof fc comes directly from evaluating the integral in Eq.12 as the normalized, response to a jump in the laser isg(t) = − exp(−fct). The uniform estimate for the residualof a second-order expansion of a Lorentzian function (aboutzero) is given by

∣∣�2(f

)∣∣ ≤ S(0)

(f

fc

)4

(18)

The upper bound of fc/4 limits the approximation errorto at most two orders of magnitude below the asymptoticplateau value of S(f ). A secondary byproduct of the fittingrange is that it makes our method of calibration less sensitiveto high-frequency spectral distortion (e.g., harmonic noise

and aliasing) that can make fitting a Lorentzian distributionperilous.

By Parceval’s theorem, integrating the PSD over all pos-itive frequencies yields the time-averaged, mean squareddisplacement of the particle:⟨x2

⟩= 1

π

∫ ∞

0S(f ) df. (19)

This quantity is directly proportional to the work done bythe optical trap. At thermal equilibrium, equipartition dic-tates that κ〈x2〉 = kBT (Neuman and Block 2004). Deter-mining trap stiffness with the above relation is straightfor-ward, as the ensemble-averaged quantity is given by thesquared standard deviation of the particle position in a statictrap. Knowledge of particle size and, more importantly,medium viscoelasticity are not required. Thus, equiparti-tion can be applied generally. Stiffness values generatedin this way are also presented in Table 1. Equipartitionassumes that the stochastic displacement measured is duesolely to thermal fluctuations of the particle in the trap. Weshall demonstrate that this assumption can be quite poor inexperimentally relevant circumstances.

The trap stiffness and medium resistivity measured bySIR at different laser powers are given in Table 1, alongsideparameter values obtained from fitting the power spectra toa Lorentzian and application of the equipartition theorem.Theory suggests that trap stiffness is linearly proportionalto laser power. This relationship is largely preserved overthe experimental range by the SIR method. Fitting the PSDto a Lorentzian yields a similar linear trend and equivalentparameter values.

In contrast, equipartition measures a strong nonlineardependence of trap stiffness on laser power. This is directlyattributable to low-frequency noise which augments thePSD above the physical low-frequency plateau. Equation19 suggests that this additional power makes time-averagedfluctuations appear larger so that the inferred trap stiffness

Table 1 The resistivity, R(0) = 6πηa, and trap stiffness, κ , measuredin water at different laser powers via calibration by SIR, Lorentzian fit(denoted Lor.), and equipartition (denoted equ.) demonstrate that theSIR method is at least as accurate as the typical Lorentzian fit

Power R(0) R(0), Lor. κ κ , Lor. κ , equ.

(mW) (cP μm) (cP μm) (pN/μm) (pN/μm) (pN/μm)

100 12.3 11.4 24 22 16

200 12.7 11.0 46 43 25

300 12.0 10.8 66 64 33

400 12.0 10.5 90 83 39

500 11.1 7.80 123 104 45

Conversely, equipartition underpredicts the trap stiffness and perceivesa nonlinear relationship between trap stiffness and laser power due tolow-frequency noise in the measured data

Rheol Acta

is an underprediction. The underprediction is exacerbatedat higher laser powers as the relative contribution of extra-neous noise to the PSD increases. Conversely, the errorassociated with the equipartition calibration may be pro-gressively mitigated by any combination of decreasing trapstiffness, increasing particle size, and increasing mediumviscosity. There are obvious limits on the applicability ofequipartition.

At 500 mW, the SIR and Lorentzian approaches appearto differ in the calibrated values they produce. This diver-gence is directly attributable to aliasing of high-frequencycomponents in the particle fluctuation signal. It is the short-time diffusion that dictates the resistivity in the Lorentzianfit, and limited high frequency data (f > fc) is available atthis power (Berg-Sørensen and Flyvbjerg 2004; Otto et al.2011). Although not immune to this metrological artifact,we expect the SIR technique to be less sensitive to alias-ing because it utilizes data from the low-frequency regime(f < fc) of the PSD. The stiffnesses and resistivities deter-mined by SIR and the Lorentzian fit of the PSD are in goodagreement at powers below 400 mW, indicating that ourapproach enables calibration with precision comparable tostandard techniques. The resistivity is independent of laserpower, thus its value should be constant. The standard devi-ation in resistivity measured by SIR was within 5 % of themean at all laser powers (12.0 cPμm).

Calibration in viscoelastic fluids

Three aqueous solutions of PEO (200 kDa) at 1.25, 2.5,and 5 % w/w are used as model viscoelastic media fortweezer calibration in a non-Newtonian fluid. The polymerconcentrations are up to an order of magnitude greater thanthe overlap concentration (c∗ ≈ 0.5 % w/w). At concen-trations exceeding this threshold, the fluid has significantviscoelasticity, the components of which can be measuredby microrheology (Dasgupta et al. 2002).

We conducted calibrations analogous to those performedin water for each PEO solution at a laser power of 100 mW.The results are listed in Table 2. In these experiments, thejump frequency was set to 1, 1, and 0.5 Hz, respectively, inanticipation of a slower relaxation of the particle back into

Table 2 The steady shear viscosity, R(0)/6πa, characteristic of elas-ticity f 2

c R′(0)/iκ , and trap stiffness, κ , measured in a 1.25, 2.5, and5 % w/w PEO solutions as measured via calibration by SIR

Conc. % (w/w) R(0)/6πa ηCP (cP) κ (pN/μm) f 2c R′(0)/iκ

1.25 3.6 3.8 48 0.05

2.5 8.6 9.3 44 1.8

5 39 42 23 2.3

the trap. The validity of the results is established by com-paring steady shear viscosity values, given by R(0)/6πa,obtained from the calibration, and those measured with acone and plate rheometer (TA instruments, AR 2000, 60-mm-1◦ cone). The particle radius is not known exactly fora given experiment as each uses a different colloidal probe.We use a value of a = 0.79μm, which is the mean valuereported by the manufacturer with a 5 % coefficient of vari-ation. The macro- and microrheology results differ by lessthan 8 % in all cases (Table 2), demonstrating the efficacyof our method in non-Newtonian media.

The trap becomes less stiff at higher polymer concentra-tions (Table 2) due to an increase in the solution’s refractiveindex and additional optical extinction through absorptionand scattering. These effects reduce the dielectric contrast(between particle and fluid) and incident optical power,respectively. Note also that the reported stiffness values inthe PEO solutions are all higher than that measured in water.This occurred because the laser was realigned betweenexperiments in water and PEO, increasing the power trans-mitted through the objective and, thus, the trap stiffness.These sample-dependent and instrumental effects illustratethe importance of in situ calibration with regard to makingaccurate and reproducible quantitative measurements withoptical tweezers.

Despite the higher trap stiffnesses, the corner frequencieswere markedly lower in the presence of PEO, reaching val-ues of 150, 56, and 6.3 Hz for the 1.25, 2.5, and 5 % w/wsolutions, respectively. This reflects a reduction in particlemobility and is visually apparent from the PSD’s shift tolower frequencies (Fig. 5).

Fig. 5 The PSDs for a PMMA particle trapped in three aqueous PEOsolutions. A laser power of 100 mW was employed and the PSD waswindowed over 1-s periods. The dashed lines correspond to the plateauvalues predicted by the SIR calibration

Rheol Acta

Optical tweezer microrheology

Once the QPD sensitivity and trap stiffness are known, theoptical tweezers can be employed to measure a material’sviscoelasticity in a variety of ways. The components of thecomplex modulus are often discerned piecewise by oscillat-ing the probe at a number of discrete frequencies. However,the calibration scheme is itself a form of active microrheol-ogy, as evidenced by the viscosity measurements describedearlier. This was also demonstrated previously by Preeceet al.; however, their impulse measurements were entirelyseparate from their calibration (Preece et al. 2011). Not onlythat we have calibrated the laser trap, but we have alsoextracted the linear viscoelasticity of the solvent.

Visual inspection of the TR curves provides qualitativeevidence of non-Newtonian behavior (see Fig. 6). While therelaxation of the particle after a jump in the laser exhibits anexponential decay at a single rate in a Newtonian fluid, inthe PEO solutions, at least two distinct relaxation regimesare observed. Physically, the initial relaxation occurs at arate proportional to the trap stiffness and inversely pro-portional to a characteristic viscous modulus. Later in theprocess, when the particle has nearly reached a minimum inthe trapping potential, there remain elastic modes of defor-mation in the solvent itself that must relax. The decay ratein this latter portion of the curve is set by the ratio of char-acteristic elastic and viscous moduli for the solvent. On alog-linear plot of g(t), at least two linear regimes are appar-ent in the PEO solutions, reflecting the above processes.This is in contrast to the TR in water which shows onlya single relaxation rate. The existence of multiple relax-

Fig. 6 The TR functions, g(t) in PEO solutions at a laser powerof 100 mW. The response in PEO shows two exponential decays:one at a high rate for tfc/(2π) < 0.25 and one at a slow ratefor tfc/(2π) > 0.25. Two dashed lines indicating these exponentialdecays are shown to guide the eye. This is a clear evidence of thenon-Newtonian character of the PEO solutions

ation regimes is most apparent in the 5 % solution (as oneshould expect) but is more difficult to discern visually at thetwo lower concentrations. However, the existence of elasticmodes at all three concentrations is more readily appar-ent from the a characteristic elasticity values, f 2

c R′(0)/iκ ,presented in Table 2. This quantity is identically zero fora Newtonian fluid and was measured as less than 10−2

in water at all laser powers. In contrast, the characteris-tic elasticity for the 1.25 % PEO solution is O(10−2) andO(1) for the more concentrated solutions. The results inwater provide an effective signal-to-noise metric for themicrorheometer with regard to rheological measurementsbased on the probe’s impulse response.

Various methods exist for converting the TR to thecomplex modulus of the material. These methods are allapplicable over a limited frequency range because the trapbehaves like a low-pass filter, incapable of transmittingdrive forces at frequencies much larger than its own charac-teristic frequency, fc. The relevant dynamic range coincideswith the plateau and transition regions in the PSD (Fig.5). It is crucial to note that this mechanical limitation doesnot curtail the instrument’s dynamic range in the linearrheological regime. While it may not be capable of driv-ing the probe’s motion at high frequency, the trap can stilltransduce any additional forces experienced by the probe.On time scales much shorter than fc, the particle’s trans-lational dynamics are independent of the trap and whollysubject to the material’s thermal fluctuations. Consequently,the medium’s complex modulus can be elucidated at highfrequency from the decaying portion of the PSD using thesame methods applied to transform the TR (Preece et al.2011). Hence, measurements with a static optical trap areakin to a single-particle version of photon correlation spec-troscopy (dynamic light scattering), a staple technique inpassive microrheology.

We use the method of Preece et al. to calculate g(f )

from the TR (Preece et al. 2011). Subsequently, Eq. 11and the Fourier transformation of the TR curve is used tocalculate the complex modulus (G(f ) = if R(f )/6πa).The complex moduli for water and the 5 % PEG solutionmeasured with a laser power of 100 mW are presented inFig. 7 along with the results from several oscillatory mea-surements. In this plot, the frequency is normalized on thecorner frequency and the modulus is normalized on κ/6πa.

The complex modulus may be partitioned as G(f ) =G′(f ) + iG′′(f ), where, G′(f ) and G′′(f ) are the storageand loss moduli, respectively. In water, the elastic modulusnever reaches statistically nor physically significant levels,and the linear growth of the viscous modulus is consis-tent for a Newtonian fluid. The viscous modulus for PEOalso scales with f and is two orders of magnitude greaterthan G′(f ) at low frequencies, indicating that, over longtimes, the polymer solution relaxes viscously. A significant

Rheol Acta

Fig. 7 Storage and loss moduli for the 5 % PEO solution (red cir-cles) and water (purple triangles) measured by impulse response at alaser power of 100 mW. Moduli values for PEO were also obtainedby oscillating the probe at discrete frequencies (black squares). Forscale, at 15 Hz (f/fc = 2.5), we measure the values of the moduli:G′ ≈ 0.95 Pa and G′′ ≈ 2.9 Pa

degree of scatter is present in the elastic modulus obtainedvia impulse response. Plotting the data on a linear ordinaterevealed a symmetric divergence from the line G′(f ) =0 with increasing frequency. This obscures any quantita-tive insight into G′(f ) but is fundamentally different fromthe results obtained in water, suggesting that it is incitedby the material’s elasticity. It is important to note that wehave proposed using the impulse response for calibrationand demonstrated its accuracy in this regard. Rheologicalmeasurements can be performed in a variety of ways withthe optical tweezers so long as SIR fulfills this primarytask. This is demonstrated by the oscillatory measurementsincluded in Fig. 7. These data show the elastic modulusscales with f 2 at low frequencies. Comparable magnitudesfor the moduli and a similar power law exponent for theelastic modulus was measured previously in 200-kDa PEOsolutions at roughly the same concentration (Dasgupta et al.2002).

Operating regimes of a microrheometer

Over what range of material properties and experimentalparameters does a laser tweezer microrheometer generatemeaningful measurements?

While an experiment may be run for as long as the sampleis viable, the QPD must be sampled by a data acquisitionsystem with sufficient frequency to detect the relaxation ofthe particle in the trap. The sampling frequency must be atleast an order of magnitude higher than the corner frequencygiven by κ/R(0). For a given material (i.e., R(0)), the corner

frequency may be reduced by decreasing the laser powerand, thus, the trap stiffness.

The trap stiffness has a lower limit. The particle mustremain trapped for the entirety of the experiment. Thisnecessitates that κλ2/kT � 1 where λ, the laser wave-length, is a suitable proxy for the linear extent of thetrapping potential. This is almost always the case with lasertweezers because the trap’s lateral extent is diffraction-limited. As a consequence, the minimum reasonable cornerfrequency is on the order of 1 Hz in water. The laser powermay be adjusted as needed to accommodate hardware witha sampling rate faster than this. The upper limit on the trapstiffness is set in part by the heating caused by absorptionof the laser in the sample. The heating rate in water for anNd–YAG laser is approximately 8 ◦C/W and may be higherin other materials (Peterman et al. 2003). Putting too muchpower though an objective may damage the optics as well.There are other restrictions on the maximum trap stiffness.

Part of the SIR calibration method depends on deter-mining the PSD of the probe’s thermal motion—this is ameasurement of noise. In the PSDs presented herein, a low-frequency noise signal is obvious. As already mentioned,this may arise from any number of sources and is difficult ifnot impossible to remove entirely. Instead, consider that thisextraneous noise decays faster than the signal generated bythe particle. At a critical frequency, f ∗, this low-frequencynoise is eliminated, and analysis of the data in terms ofthe particle’s Brownian motion is justified. If the contri-bution to the PSD from the low-frequency noise is A/f ,with A setting the magnitude, then its intersection with theplateau of the Brownian contribution to the PSD, that criticalfrequency, scales as

f ∗ = Aκ2

2kT R(0). (20)

In order to make the SIR measurement accurately, werequire that f ∗ is much smaller than the corner frequency.This ensures that there is a large portion of the plateau in thePSD to fit for the trap stiffness—necessarily:

Aκ

kT� 1. (21)

The quantity kT /κ may be identified from equipartition asthe mean squared fluctuation in particle position due solelyto thermal motion. We may conclude that A characterizesthe mean squared fluctuation due to extraneous noise andthat this must be smaller than the Brownian signal. Increas-ing trap stiffness and, thus, localizing the particle willeventually violate this criterion. Then, the PSD becomes sat-urated with extraneous noise. For SIR calibration, kT /λ2 �κ � kT /A.

Additionally, the QPD which detects the motion of theparticle has a noise floor of ∼ 2 mV, which, given the typical

Rheol Acta

QPD sensitivity of 100 nm/V, means instantaneous displace-ments of the probe smaller than 2 A could not be measuredwith statistical significance. The maximum useful displace-ment measurable is set by the linear range of the QPD. Wefind that this is typically around 300 nm. The product ofdisplacements with the trap stiffness set upper and lowerbounds on the forces than can be measured. A range from2 fN to 100 pN is theoretically possible.

Interestingly, the viscoelastic moduli play no direct rolein defining these measurement limits. It is a dynamicalquantity governing only how quickly relaxation happens.We have demonstrated the ability to measure the viscositywith high accuracy over almost two decades in magnitude(1–40 cP). For high viscosity materials one, merely needwait longer to perform the calibration.

Conclusions

We have developed a simple two-step procedure, sequen-tial impulse response, for calibrating the trapping force andthe detector sensitivity in a material of unknown viscoelas-ticity. First, the passive diffusion of the particle within thetrap is measured in the static portion of the procedure. Thenthe transient relaxation of the particle in response to a jumpin the traps position is measured repeatedly in the dynamicportion of the procedure.

These two measurements encode all the information re-quired to extract the relevant calibration parameters. There-fore, this method may be applied equally well in Newtonianand non-Newtonian fluids. Interestingly, the determinationof trap stiffness and detector sensitivity requires no fore-knowledge of the solvent’s viscoelastic moduli nor an expli-cit form of its resistivity, which depends on both viscoelas-ticity and probe geometry. This procedure may be used toeasily characterize the trapping of nonspherical particles.

Any number of experiments may be undertaken after cal-ibration. The same optical tweezers allow for quantitativeexamination of the solvent rheology far from equilibriumvia active and nonlinear microrheology (Meyer et al. 2006).Measurements of colloidal interaction forces can be madeeasily by trapping more particles (Park and Furst 2011;Pantina and Furst 2004). Even the hydrodynamic inter-actions among particles in non-Newtonian media can beprobed by inducing strong motion among trapped particles(Lele et al. 2011). Calibration of the trapping and detectionequipment and measurement of linear viscoelasticity are butan overture. The concert of rheology and colloid sciencefollows.

Acknowledgements Funding from the International Flavors andFragrances and NASA (grant no. NNX10AE44G) is gratefullyacknowledged. The authors thank Mark Pancyzk for his assistance.

References

MacKintosh FC, Schmidt CF (1999) Microrheology. Curr Opin Col-loid Interface Sci 4:300–307

Squires TM, Mason TG (2010) Fluid mechanics of microrheology.Annu Rev Fluid Mech 42:413–438

Furst EM (2005) Applications of laser tweezers in complex fluidrheology. Curr Opin Colloid Interface Sci 1:79–86

Brau RR, Ferrer JM, Lee H, Castro CE, Tam BK, Tarsa PB,Matsudaira P, Boyce MC, Kamm RD, Lang MJ (2007) Passive andactive microrheology with optical tweezers. J Optics A 9:S103–S112

Mizuno D, Head DA, MacKintosh FC, Schmidt CF (2008) Activeand passive microrheology in equilibrium and non-equilibriumsystems. Macromolecules 41:7194–7207

Park BJ, Furst EM (2011) Attractive interactions between colloids atthe oil-water interface. Soft Matter 7:7676–7682

Lele PP, Swan JW, Brady JF, Wagner NJ, Furst EM (2011) Col-loidal diffusion and hydrodynamic screening near boundaries. SoftMatter 7:6844–6852

Neuman KC, Block SM (2004) Optical trapping. Rev Sci Instrum75:2787–2809

Gittes F, Schmidt C (1998) Interference model for back-focal-planedisplacement detection in optical tweezers. Opt Lett 23:7–9

Berg-Sørensen K, Oddershede LB, Florin EL, Flyvbjerg H (2003)Unintended filtering in a typical photodiode detection system foroptical tweezers. J Appl Phys 93:3167–3176

Fischer M, Richardson AC, Nader S, Reihani S, Oddershede LB, Berg-Sørensen K (2010) Active-passive calibration of optical tweezersin viscoelastic media. Rev Sci Instrum 81:015103

Le Gall A, Perronet K, Dulin K, Villing A, Bouyer P, Visscher K,Westbrook N (2010) Simultaneous calibration of optical tweez-ers spring constant and position detection. Opt Express 18:26469–26474

Berg-Sørensen K, Flyvbjerg H (2004) Power spectrum analysis foroptical tweezers. Rev Sci Instrum 75:594–612

Preece D, Warren R, Evans RML, Gibson GM, Padgett MJ, CooperJM, Tassieri M (2011) Optical tweezers: wideband microrheology.J Optics 13:044022

Pesce G, De Luca AC, Rusciano G, Netti PA, Fusco S, Sasso A(2009) Microrheology of complex fluids using optical tweez-ers: a comparison with macrorheological measurements. J Opt A11:034016

Otto O, Gornall JL, Stober G, Czerwinski F, Seidel R, Keyser UF(2011) High-speed video-based tracking of optically trapped col-loids. J Optics 13:044011

Dasgupta BR, Tee SY, Crocker JC, Frisken BJ, Weitz DA (2002)Microrheology of polyethylene oxide using diffusing wave spec-troscopy and single scattering. Phys Rev E 5:051505

Peterman EJG, Gittes F, Schmidt CF (2003) Laser-induced heating inoptical traps. J Biophys 84:1308–1316

Meyer A, Marshall A, Bush BG, Furst EM (2006) Laser tweezermicrorheology of a colloidal suspension. J Rheol 50:77–93

Pantina JP, Furst EM (2004) Directed assembly and rupture mechanicsof colloidal aggregates. Langmuir 20(10):3940–3946