Active microrheology to determine viscoelastic parameters of Stokes-Oldroyd B fluids using optical tweezers
AActive microrheology to determine viscoelasticparameters of Stokes-Oldroyd B fluids using opticaltweezers
Shuvojit Paul
Indian Institute of Science Education and Research, Kolkata
Avijit Kundu
Indian Institute of Science Education and Research, Kolkata
Ayan Banerjee
Indian Institute of Science Education and Research, KolkataE-mail: [email protected]
Abstract.
We use active microrheology to determine the frequency dependentmoduli of a linear viscoelastic fluid in terms of the polymer time constant ( λ ), and thepolymer ( µ p ) and solvent viscosity ( µ s ), respectively. We measure these parametersfrom the response function of an optically trapped Brownian probe in the fluid underan external perturbation, and at different dilutions of the viscoelastic component in thefluid. This is an improvement over bulk microrheology measurements in viscoelasticStokes-Oldroyd B fluids which determine the complex elastic modulus G ( ω ) of thefluid, but do not, however, reveal the characteristics of the polymer chains and theNewtonian solvent of the complex fluid individually. In a recent work [Paul et al .,2018 J. Phys. Condens. Matter µ p ) and ( µ s ), which we now extend to account for an external sinusoidal forceapplied to the probe particle. We measure λ , µ p , and µ s experimentally, and comparewith existing the λ values in the literature for the same fluid at some of the dilutionlevels, and obtain good agreement. Further, we use these parameters to calculate thecomplex elastic modulus of the fluid again at certain dilutions and verify successfullywith existing data. This establishes our method as an alternate approach in the activemicrorheology of complex fluids which should reveal information about the compositionof such fluids in significantly greater detail and high signal to noise. Keywords :Viscoelastic, Stokes-Oldroyd B, Optical Tweezer, Microrheology, ViscoelasticParameters a r X i v : . [ c ond - m a t . s o f t ] N ov
1. Introduction
The fundamental difference between liquids and solids is their response under appliedshear strain - while solids store energy and thus are elastic, liquids dissipate energy andare therefore viscous in nature. However, fluids ranging from cytoplasm to ketchup,store and dissipate mechanical energy in relative proportions depending on frequency.Therefore, they are called viscoelastic. There exists a strong interest in the scientificcommunity to understand and measure the parameters of viscoelastic fluids mainlybecause, the biological entities which sustain life are viscoelastic [1, 2, 3, 4, 5, 6, 7].The rheological properties of such fluids are often parameterized in terms of a frequencydependent complex elastic modulus G ∗ ( ω ) whose real part G (cid:48) ( ω ) remains in phase withthe applied strain and represents the storage of energy (elastic part), while the imaginarypart G (cid:48)(cid:48) ( ω ) remains out of phase, and represents the loss (viscous part) of energy in thesystem [7, 8, 9, 10]. The complex dynamic viscosity is given by η ( ω ) = G ∗ ( ω ) / ( − iω ).Typically, the bulk rheological properties of a viscoelastic material is measured byanalyzing its response when the entire sample is subject to an external strain. Therefore,the local heterogeneity in the sample remains unexamined [11, 12, 13]. Additionally,this method commonly requires ∼ mL of samples which may limit its use for expensiveor scarce samples, such as biological fluids. The invention of optical tweezers in 1986 byAshkin and colleagues [14], has facilitated ’microrheology’ (rheology in the micrometerscale) with ∼ µ L of samples and the above-mentioned issues have been overcome [15, 16].In this method, typically, the Brownian motion (passive microrheology) or the motionunder external perturbation (active microrheology) of a micron sized trapped particleinside a fluid is studied to extract the frequency dependent viscoelastic parameters G (cid:48) ( ω )and G (cid:48)(cid:48) ( ω ). Active microrheology understandably provides enhanced capabilities andwider parameter space of rheological measurements along with better signal-to-noiseratio (S/N) over the passive technique [17, 18].To get deeper insight into the sample property, different models describing a linearviscoelastic fluid have been developed. Foremost among these, is the Maxwell model[19, 20] which has been further developed into the generalized Maxwell model or Jeffreys’model [21, 22]. The high degree of simplification [23, 24, 25] used in the Maxwellmodel ease out calculations, but the model sometimes fails to interpret experimentalresults. It has been shown that at least the Jeffrey’s model is required to explain andunderstand experimental results in detail [26]. Both of these models can provide thestress and shear-strain relation for a linear viscoelastic fluid in terms of the fundamentalparameters of the fluid. For example, the Maxwell model describes a time constant τ M which marks a transition from the high frequency elastic nature of the sample tothe low frequency viscous regime, while the Jeffrey’s model contains a zero-frequencyviscosity η and a correction term as the background viscosity η ∞ [27]. However, thesemodels are based on the bulk properties of the viscoelastic fluid and do not provideany information about its basic constituents. To address this issue, we have shown ina recent work that a viscoelastic fluid can be understood as a viscous solvent whichcontains a polymer network mixed with it. We have demonstrated that the backgroundviscosity η ∞ is nothing but the solvent contribution to the zero-frequency viscosity, while η is the polymer contribution to it, while the Maxwell time constant is basically thepolymer time constant [28]. We obtained this understanding by linearizing the Stokes-Oldroyd B equations for small perturbations and for low Weissenberg number. Clearly,this approach links the overall rheological behavior of the fluid with the characteristicsof its constituents and provides greater acuity in measurement and understanding ofviscoelasticity itself.In this paper, we measure for the first time, the polymer and solvent contributionsto the viscoelasticity of a linear viscoelastic fluid having a single time constant. Thus,we experimentally determine the phase response of a micron sized spherical particleconfined in a harmonic potential in such a Stokes-Oldryod B fluid under an externalperturbation. First, we solve the equation of motion of a trapped particle under externalperturbation in a fluid as described in the recent work [28]. Then, we fit the expressionof the phase response to experimentally measured data to extract the parameter values.Typically, in the passive microrheology technique, a generalized Stokes-Einstein relationis employed to convert the mean-squared displacement (MSD) of the probe into thecomplex elastic modulus G ∗ ( ω ). This process involves a fourier transform of the MSD,which is rather non-trivial in practice, given a finite set of data points over a finitetime domain [10]. Understandably, viscoelastic fluids with very low concentration of thepolymer network, can be very easily analyzed by this simple method. In addition, thismethod involves measuring the MSD which is obtained from the amplitude of Brownianmotion of a probe particle. However, we have shown recently that phase measurementusing a lock-in is more sensitive and accurate than amplitude measurement [18], whichis not unexpected since the amplitude of a signal gets more effected by noises thanthe phase. Further, the measurement from the phase does not require the conversionof the signal into real displacement units, so that errors involving in determining thecalibration factor can be avoided (which, incidentally, is significantly affected by detectorelectronics). For consistency check, we have applied this phase-measurement basedtechnique to normal water and obtained good agreement in our determination of thesolvent viscosity. Further, we have proceeded to measure the viscoelastic parametersfor samples of Polyacrylamide (PAM) to water solutions at different dilution levels. Weobserve that our results are in good agreement with that reported recently [29], whichhave been performed for relatively low polymer concentration solutions. For solutionsof higher polymer concentrations, however, the measured polymer contributions to theviscosities are not satisfactory. We believe this to be due to the inherent ineffectivenessof our model in dealing with the non-linear nature of viscoelasticity or the additionalcomplexity resulting in the superposition of several time constants and other parametersthat the high concentration of polymer would induce in a fluid [8]. For linear viscoelasticfluids and for low-concentrations, our work opens a new approach in microrheology andcan be used very extensively due to its simple methodology and ease-of-use.
2. Theory
The equation of motion describing the trajectory of a spherical particle of mass m confined in a harmonic potential of force constant k in a linear viscoelastic fluid in theCartesian co-ordinate system (we choose x here) is given by m ¨ x ( t ) = − (cid:90) t −∞ γ ( t − t (cid:48) ) ˙ x ( t (cid:48) ) dt (cid:48) − k [ x ( t ) − x ( t )] + ξ ( t ) (1)where, the integral term on the right-hand side incorporates a generalized time-dependent memory kernel, and γ ( t ), represents damping by the fluid, so that it can betermed as the time dependent friction coefficient. x ( t ) and x ( t ) are the instantaneouspositions of the particle and the potential minimum, respectively, and ξ ( t ) is theGaussian-distributed correlated thermal noise due to the random collisions of the fluidmolecules with the particle which leads to the Brownian motion of the particle. Thecorrelation of the noise is given by (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = 2 k B T γ ( t − t (cid:48) ) where k B is the Bolzmannconstant and T is the temperature. Due to the negligible mass of the trapped probeparticle and the fact that we are at low Raynold’s number, the momentum relaxationtime scale and the vorticity time scale are negligible compared to typical experimentaltime scales. Thus, we neglect the inertial term from Eq. 1 and average over thenoise since we are basically interested in the response function of the particle underexternal perturbation. Therefore, the equation of motion in the frequency domain canbe effectively written as − iωγ ( ω ) x ( ω ) + kx ( ω ) = kx ( ω ) (2)Now, γ ( ω ) of a Stokes-Oldroyd B fluid is related to the polymer time constant λ , thepolymer and the solvent contribution to the viscosity µ p and µ s , respectively, as [28] γ ( ω ) = 6 πµ s a (cid:18) µ r − iωλ + 1 (cid:19) (3)where a is the radius of the trapped particle and µ r = µ p /µ s . Substituting Eq. 3 inEq. 2 and calculating the phase of the response of the particle we getΦ ( ω ) = tan − µ r λ ω + ω kλ γ + (cid:16) kγ + µ r λ (cid:17) ω (4)where γ = 6 πµ s a . Therefore, it is possible to fit the experimentally measured phasewith Eq. 4 to infer the characteristic parameters of the concerned fluid. Later on, theseparameters can be employed to obtain the complex shear modulus of the fluid which isgiven by G ∗ ( ω ) = − iωγ ( ω ) / πa for a spherical probe particle [28] .
3. Experimental Details
We perform the experiments using an optical tweezers built around an invertedmicroscope (Zeiss Axiovert.A1 Observer) with an objective lens (Zeiss PlanApo 100x,1.4 numerical aperture) tightly focusing a laser beam of wavelength 1064 nm into the P h a s e ( D e g r ee ) -21 -13 -5 G ' & G '' ( P a ) G' G'' 0.0062~ Slope m p = 6.1×10 -7 Pa.s l = 2.2×10 -14 s Data Fit Figure 1.
Phase response of the spherical probe of radius 1 . µ m in water as afunction of driving frequency along with a fit to the data using Eq. 4. The measuredtrap stiffness is 52(4) µ N/m. In the inset, G (cid:48) ( f ) and G (cid:48)(cid:48) ( f ) have been plotted againstfrequency - these have been calculated theoretically using the extracted parametervalues of the fluid. The loss part G (cid:48)(cid:48) ( f ) increases linearly with frequency having a slopof 2 πµ s = 0 . G (cid:48)(cid:48) ( f )is close to zero. λ and µ p also tend to zero which imply the effective phase responseto be φ ( f ) = tan − (2 πγ f /k ). This is also is valid for a viscous fluid. P h a s e ( D e g r ee ) -4 -2 G ' & G '' ( P a ) G' G''
Data Fit m p = 9.2×10 -4 Pa.s l = 0.00127 s Figure 2.
Phase response of the probe particle of radius 1 . µ m in 0.008% w/w PAMto water solution with driving frequency along with a fit to the data using Eq. 4.The measured trap stiffness is 48(3) µ N/m. In the inset, G (cid:48) ( f ) and G (cid:48)(cid:48) ( f ) have beenplotted against frequency which has been calculated theoretically using the extractedparameter values of the fluid. sample. For detecting the displacement of trapped particles, we employ a co-propagatinglaser of wavelength 780 nm. A balanced detection system [30, 18, 31] placed at theback-focal plane of the objective detects the back-scattered light from the trappedprobe particle to track its position. We modulate the trapping laser beam at differentfrequencies by a piezo-mirror placed at the conjugate plane of the objective focal planeand keep the detection beam fixed. Simultaneously, we use a CCD camera to imagethe trapped particle. Our sample is inserted into a sealed sample chamber which isprepared by attaching a glass slide to a cover slip by double-sided tape so that thedimensions become around 20 × × . M w = (5 − × gm/mol, Sigma-Aldrich) and trapped single spherical polystyrene probe particles ofradius 1 . µm around 30 µm away from the nearest wall to get rid of surface effects.We modulate the trap center sinusoidally by the piezo-mirror with an amplitude of 110nm at different frequencies and record the response for 60 seconds at each frequencyby a data acquisition card (NI USB-6356). Simultaneously, the data is fed into a lock-in amplifier (Standford Research, SR830) and averaged over the same time duration tomeasure the relative phase of the response of the particle with respect to the modulation.In the absence of the modulation, we have recorded the Brownian motion of the particleto calculate the trap stiffness.
4. Results and discussions
In equilibrium, the stiffness of the optical trap can be measured using the equipartitiontheorem since the latter is independent of the rheological property of the sample.According to this theorem, the trap stiffness in our system is given by k = k B T / (cid:104) ( x − (cid:104) x (cid:105) ) (cid:105) . After we determine the trap stiffness, we determine the fluidparameters for different concentrations by plotting the measured phase of the proberesponse as a function of driving frequency, and fitting the data to Eq. 4. The fitparameters then yield the parameters of the fluid. To check for the efficacy of ourtechnique, we have first performed the measurement for pure water and obtainedgood agreement. The results are demonstrated in Fig. 1 along with the correspondingstorage ( G (cid:48) ) and loss ( G (cid:48)(cid:48) ) moduli in the inset. The time constant λ and the polymercontribution of viscosity µ p are both close to zero, and the solvent contribution ofviscosity is 0 . ± . φ ( ω ) = tan − ( γ ω/k ), which is indeed thephase response of a particle trapped in a pure viscous fluid. The storage modulusis almost zero whereas the loss modulus increases with frequency having slope 0 . π × µ s , [32] thus corresponding to a purely viscous sample. In Fig.2, we show a typical phase response of a linear viscoelastic fluid (0 . G (cid:48) ) ismuch greater ( ∼ ). It is important to point out that for both the cases we keptthe trap stiffness fixed. We then evaluated the parameter values for PAM to watersolutions of different concentrations, which are shown in Table 1. The evaluated solventcontribution of viscosity is µ s = 0 . ± . Table 1.
Extracted parameters with varying PAM concentrations in water. Thestiffness of the trap has been kept fixed at k = 48(3) µ N/m over the measurements.The extracted solvent viscosity from all the measurements is µ s = 0 . concentration (% w/w) λ (s) µ p (Pa.s)0.002 0.00031(4) 0.00036(5)0.004 0.00053(6) 0.00032(4)0.006 0.0008(1) 0.00047(7)0.008 0.0013(1) 0.00083(4)0.03 0.0028(2) 0.0012(1)0.06 0.0051(5) 0.0015(1)0.1 0.0057(8) 0.0014(2)0.5 0.0033(5) 0.0015(2)1 0.006(1) 0.004(1) time constant increases with the concentration of the solution and almost saturates after0 . λ are consistent with a recent work (Fig. 5 of Ref. [29])- for ease of comparison, we have juxtaposed their data with ours in Fig. 3. Clearly,our measurements follow the same trend in the variation of λ against frequency asreported in this paper. On the other hand, the polymer contribution of viscosityalso increases with the PAM concentration in the solution as expected. The smalldeviations of our measurements from the reported values can be due to the differencesin experimental conditions including local temperature, the molecular weight of PAM,electronic noise, etc. Furthermore, our measurements are really localised involving avery small region of the fluid, whereas the reported measurements are bulk in nature,so that differences may appear due to local temperature fluctuations, density variationsdue to inhomogeneous mixing, and other local effects. Further, it is clear from Table 1that the measured polymer contribution to the viscosity µ p does not appear reliable forhigher concentrations, with the change of µ p being rather small with large change ofconcentration. This basically suggests that at these levels of viscoelasticity, where thenature of the viscoelastic response may become non-linear to applied strain, or additionaltime constants may appear [8], our theory may have limitations since it only accountsfor linear viscoelastic response and a single time constant. It is thus likely that we aremeasuring a superposition of different time constants at higher polymer concentrations,and thereby obtaining erroneous results.
5. Conclusions
In conclusion, we have presented a simple experimental method to extract the rheologicalparameter values of a linear viscoelastic fluid using optical tweezers. Our methodemploys active microrheology, which straightaway enhances the signal to noise of themeasurements. Thus, we measure the phase response of a Brownian probe particlethat we modulate sinusoidally in an optical trap at different frequencies. The inherent l ( m s ) -3 Our result Reported in Ref. [29]
Figure 3.
Measurement of the polymer time constant λ as a function of PAM towater concentration using our technique superimposed with the values reported in theliterature [29]. The trends of the variation of λ with concentration for our values andthose reported earlier clearly appear to be similar. construction of our approach - based on linearizing the Stokes-Oldroyd B equation forviscoelastic fluids - provides for a more profound understanding about the constituentsof such fluids inasmuch that it reveals the polymer and solvent characteristics separately.Our method has a basic advantage over the most commonly used technique ofcharacterizing viscoelastic fluids from measurements of the storage and loss parameters G (cid:48) ( ω ) and G (cid:48)(cid:48) ( ω ) which involve a complex discrete fourier transformation of a finite set ofMSD data over a finite time, which we are able to avoid entirely. This Fourier transformcan be erroneous for the low fluid concentrations that are required in microrheology.Furthermore, our approach of measuring the phase using a lock-in amplifier has anobvious signal to noise advantage over techniques which measure the amplitude ofmotion of Brownian particles and are therefore much more susceptible to experimentalnoise. The phase measurement also precludes the requirement of the calibration ofthe particle displacement in real physical units for which a careful measurement of thedetector sensitivity is essential, which naturally leads to enhanced systematic errors.We test our technique on a purely viscous fluid - water, for which we obtain verygood agreement with well-known literature values, and different viscoelastic solutionsof PAM and water where the concentration of the former is varied. At low polymerconcentrations, we obtain rather reliable measurements of the time constant λ whichmatch with values in literature [29], while for increased polymer concentrations, thevalues seem to be unreliable with very small change in the measured µ p with increasingconcentration. This we attribute to the limitations of our theory in the case of non-linear viscoelastic fluids and for more complex fluids with additional time constants,which probably is the case when we increase the concentration of PAM in the solution.We also calculate G (cid:48) ( ω ) and G (cid:48)(cid:48) ( ω ) for the different fluid concentrations, and obtainexpected trends against frequency. This is an entirely new approach in microrheology,and we intend to extend our measurements to more diverse systems which still fit ourmodel in their viscoelastic response such as blood or plasma (and biological fluids, ingeneral - they being weakly viscoelastic) - where the accuracy of the technique may alsorender it as a useful diagnostic tool by comparing the viscoelastic parameters in infectedand normal conditions. We are presently commencing these experiments.
6. Acknowledgments
This work was supported by IISER Kolkata, an autonomous teaching and researchinstitute supported by the Ministry of Human Resource Development, Govt. of India.
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