Exploiting the color of Brownian motion for high-frequency ... - UNED

Exploiting the color of Brownian motion for high-frequency microrheology of Newtonian fluids Pablo Dom´ınguez-Garc´ıaa,b , Flavio M. Morb , L´aszl´o Forr´ob and Sylvia Jeneyb a Dep.

F´ısica de Materiales, Universidad Nacional de Educaci´on a Distancia (UNED), Madrid 28040, Spain; b Laboratory of Physics of Complex Matter, Ecole Polytechnique F´ ed´erale de Lausanne (EPFL), 1015 Lausanne, Switzerland ABSTRACT

Einstein’s stochastic description of the random movement of small objects in a fluid, i.e. Brownian motion, reveals to be quite different, when observed on short timescales. The limitations of Einstein’s theory with respect to particle inertia and hydrodynamic memory yield to the apparition of a colored frequency-dependent component in the spectrum of the thermal forces, which is called “the color of Brownian motion”. The knowledge of the characteristic timescales of the motion of a trapped microsphere motion in a Newtonian fluid allowed to develop a high-resolution calibration method for optical interferometry. Well-calibrated correlation quantities, such as the mean square displacement or the velocity autocorrelation function, permit to study the mechanical properties of fluids at high frequencies. These properties are estimated by microrheological calculations based on the theoretical relations between the complex mobility of the beads and the rheological properties of a complex fluid. Keywords: microrheology, Brownian motion, optical tweezers, hydrodynamic memory

1. INTRODUCTION One of the applications of understanding the physics of the random motion of a small spherical probe in a fluid, i.e. Brownian motion,1 is to estimate the mechanical properties of the liquid, which surrounds the bead. The field of science using that methodology received the name of microrheology2 and has been very useful in the last few decades for the study of the thermo-physical properties of complex fluids.3 Microrheology employs several different experimental modes and is constantly improved with new techniques and methods.4–6 We use an experimental set-up composed of an optical trap7 and an interferometric position detection system,8 which allows to measure the position of a bead immersed in a fluid with nanometer accuracy. The optical trap keeps the particle located in the middle of the sample chamber, which is essential to avoid effects caused by confinement through the presence of a surface.9, 10 However, it introduces an external trapping force, which modifies the dynamics of the bead.11 This experimental technique reaches the short-time scale down to 1µs12 and, therefore, has the capacity to provide rheological measurements in the interval 10-106 Hz, i.e. high-frequency micro-rheology.13 By exploiting hydrodynamic memory effects in Brownian motion, which dominate at short times, a calibration methodology of the signal provided by the interferometry set-up has been recently developed.14 Here, we apply this calibration methodology to quantify the Brownian movement of a melamine resin microsphere immersed in two different Newtonian fluids, water and acetone, thereby obtaining the optical trap spring constant and the Newtonian viscosity. From well-calibrated correlation quantities, we then use standard microrheological methods to further derive the mechanical properties of both fluids. Finally, we discuss limitations of microrheological calculations and new advances in theoretical microrheology to extract high-frequency mechanical properties from the short-time Brownian motion of a microsphere. Further author information: (Send correspondence to S.J.) S.J.: E-mail: sylvia.jeney@epfl.ch

Optical Trapping and Optical Micromanipulation X, edited by Kishan Dholakia, Gabriel C. Spalding, Proc. of SPIE Vol. 8810, 881015 · © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2024849

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2. METHODOLOGY The digital signal provided by the interferometric detection of the position fluctuations of an optically trapped bead immersed in a Newtonian fluid is measured in volts and has to be converted into a distance measurement. This calibration methodology yields a calibration factor β in units of µm/V and is based in the determination of experimental characteristic time scales,14 instead of the more conventional way of fitting the experimental data to a theoretical expression.15 In principle, the analytical solution of the Langevin equation16 for an optically trapped spherical particle with no-slip boundaries in a Newtonian fluid with hydrodynamic effects17 only depends numerically on three timescales τf , τp∗ and τk . They are defined as: τf ≡

m∗ ρf a2 ∗ 6πηa ; τp ≡ ; τk ≡ η 6πηa k

(1)

where a is the bead radius, η the viscosity of the fluid, ρf the density of the fluid and m∗ ≡ mp + mf /2 an effective mass, which modifies the mass of the particle mp by mf , the mass of the displaced fluid. The first timescale, τf , is the time needed by the fluid vortex to propagate itself over one particle radius. The second one, τp∗ , is an inertial time scale. Finally, τk is the ratio between the Stokes friction coefficient and the the Hooke-like optical trap constant, k. By knowing the characteristics of the beads and the fluid, such as a, ρf and ρp , this calibration method allows to infer the timescales and, in addition, yields the values for η and k. To validate whether the numerical values provided by the calibration are correct, we calculate the mechanical properties and the viscosity from the calibrated means square displacement (MSD) using the Mason-Weitz method. This well-established microrheological methodology connects the complex mobility of a Brownian particle to the rheological properties of the surrounding fluid through the Generalized Stokes-Einstein relation (GSER)18 and Mason’s approximation.19 Open-source routines are available to the microrheological community to calculate the mechanical properties by means of these methods.20, 21 The mechanical properties are basically defined by the complex viscosity η ∗ (ω), which is related to the complex shear modulus through G∗ (ω) = iωη ∗ (ω) = G′ (w) + iG′′ (ω), where ω is the frequency of an oscillatory strain and G′ and G′′ are the elastic and loss modulus.22 The viscosity is thus calculated by: √ η(ω) ω = G′ (ω)2 + G′′ (ω)2 (2) which can be approximated to η ∼ = G′′ (ω)/ω for a Newtonian fluid. Therefore, the GSER relates the complex mobility of the particle and the rheological properties of the fluid by replacing the Newtonian viscosity with the complex viscosity by η → G∗ (ω)/(iω) through: G∗ (ω) =

kB T iωπa MSD(ω)

(3)

which is similar to the Stokes-Einstein equation for the diffusion of a particle in a fluid. Here MSD(ω) represents the one-sided Fourier transform of the MSD, which is calculated using Mason’s approximation.

3. RESULTS AND DISCUSSION Fig. 1(a) shows an example of a calibrated experimental velocity autocorrelation function, |VAF(t)| ≡ |⟨v(t)v(0)⟩|, in absolute values, for an experiment with a trapped particle of melamine resin with diameter d = 2.94 µm and density ρp = 1570 kg/m3 in acetone (ρf = 790 kg/m3 ) at T = 21◦ C. The red circles are experimental data filtered by logarithmic blocking, whereas the gray line, only visible at the end of the graph, represents the data filtered using forward and reverse filtering, as described in.23 The black line is the theoretical curve obtained from the calibration method explained above. The two minima, appearing as two “spikes” in the theoretical |VAF(t)| correspond to the point where VAF(t) crosses the zero-line. They are related by the interplay between hydrodynamic memory and optical trapping. The first zero-crossing follows the well-known algebraic power-law24, 25 in the form of ∼ t−3/2 . This tail is perfectly resolved in the experimental data, but very short due to the initial ballistic regime26 of the particle,

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Acetone, d = 2.94

(a)

m, k = 160

N/m

5

7.0x10

10

6.0x10

4

10

-9

Acetone, d = 2.94

(b)

m, k = 160

N/m

-9

-3/2

5.0x10 PSD ( m /Hz)

10

|VAF| (

2

2

2

m /s )

3

2

10

1

10

-9

Colored noise

4.0x10

3.0x10

2.0x10

-9

-9

-9

0

10

1.0x10

-9

-1

10

-6

10

-5

10

-4

10

-3

10

10

3

10

4

f (Hz)

Time t (s)

Figure 1. The color of Brownian motion of a trapped melamine resin sphere of diameter d = 2.94 µm and density ρp = 1570 kg/m3 in acetone (ρf = 790 kg/m3 ) at T = 21◦ C (a) Calibrated |VAF(t)|. The red circles show the experimental data filtered by logarithmic blocking (30 bins per decade), gray line represents the alternative filtering method and the black line give the fitted theoretical curve. (b) The color of the thermal force appears as deviations of the measured PSD (red circles) from the Lorentzian spectra (black line).

intentionally chosen to be large and heavy, as well as to the trap, k = 160 µN/m, chosen to be strong. The second zero-crossing should follow a power-lay decay in the form of ∼ t−7/2 (black line), which is, however, hidden by the noise floor of the experimental set-up. The existence of power-law behaviors in the VAF is exclusively related to the hydrodynamic memory of the fluid. One of its effects can be observed in Fig. 1(b), where the calibrated power spectral density (PSD) obtained for the same experiment is displayed. The PSD is calculated by a decomposition of the position of the bead x(t) ∫ T /2 into Fourier modes (xT (f ) = −T /2 exp(if t/2π)x(t)dt), then applying PSD(f ) ≡ limT →∞ (1/T ) ⟨|xT (f )|⟩. The semilogarithmic plot of the PSD shows a different behavior from a Lorentzian spectrum, where no hydrodynamics are considered (black line. This increase of the PSD signal in the kHz range indicates a deviation from Gaussian white noise and reflects the color of thermal noise.27 Here, the fluid’s vortex diffusion also has an influence on the spectrum of thermal forces, which then gains a colored frequency-dependent component. The most commonly used correlation quantity in microrheology is the MSD, which is defined as MSD(t) ≡ ⟨ ⟩ (x(t) − x(0))2 . In Fig. 2(a) and (b) several calibrated measurements in water and acetone using a trapped melamine bead with d = 1.88 µm at different trap stiffnesses are compared. After calibration, all curves overlap at short and intermediate times, as expected. The zero-shear viscosities obtained by our calibration method are 0.97 and 0.38 mPa.s for water and acetone, respectively. They agree very well with the values of 0.97 and 0.32 mPa.s, determined by conventional viscometry at the experiment temperature. Differences in the MSDs start to occur as soon as the optical trap dominates, which yields to a plateau with a value of MSD(∞) = 2kB T /k, according the equipartition theorem. To verify the accuracy of the measured k and β, tabled in the figure, we also calculate the normalized position autocorrelation function (NPAF) from the normalized MSD, which is independant of β. Both quantities are linked by NPAF(t) = 1 − MSD(t)/MSD(∞). NPAF follows a single exponential decay in the form of NPAF = exp(−t/τk ), when neglecting hydrodynamic memory effects. From that, it is possible to obtain the value for k, as suggested by Tassieri et al.28 Those calculations are summarized in Fig. 3(a) with the same number code as in Fig. 2(a) to identify each curve. Linear regression on the Napierian semilogarithmic plot provides k values that are quite similar to the ones summarized in Fig. 2(a): (1) τk−1 = 3.35 × 102 s−1 , hence k = 5.9 µN/m; (2) τk−1 = 9.27 × 102 s−1 , k = 16.4 µN/m; (3) τk−1 = 4.06 × 103 s−1 , k = 72 µN/m, and (4) τk−1 = 1.03 × 104 s−1 , k = 181 µN/m. Finally, we validate whether the numerical values for η provided above are correct, by calculating with the MasonWeitz method the mechanical properties and the viscosity from the calibrated MSD, displaying the weakest k in

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-2

(a)

Water, d = 1.88

m

-2

10

10

Acetone, d = 1.88

(b)

m

1

1

-3

-3

10

2

MSD ( m )

2

2

3

-4

10

4 -5

10

-6

10

-7

10

-6

10

(1)

= 12.1

(2)

= 8.8

m/V, k =18

(3)

= 5.5

m/V, k = 67

(4)

= 3.7

m/V, k = 168

-5

10

-4

10

2

1

1 MSD ( m )

10

m/V, k = 6

-3

10

m/N

3

-4

10

4 -5

10

-6

10

m/N m/N m/N

-7

10

-2

10

-6

10

t (s)

(1)

= 10.6

(2)

= 9.3

m/V, k =17

(3)

= 5.1

m/V, k = 76

(4)

= 1.6

m/V, k = 370

-5

10

-4

10

m/V, k = 5

-3

10

m/N m/N m/N m/N

-2

10

t (s)

Figure 2. Calibrated MSD(t) of a melamine resin bead with diameter d = 1.88 µm immersed in water (a) or aceton (b) at four different trap stiffnesses k.

Fig. 2 (curve number 1). The results for G′ (ω), G′′ (ω) and η(ω) in acetone and water are displayed in Fig. 3(b). The MSD for a Newtonian fluid, without the influence of an optical trap, should behave as ∼ tα where α = 1. Fig. 2 shows that this is only verified at intermediate timescales around 10−4 s. At longer times, the optical trap is dominant, whereas at lower timescales, the hydrodynamic and ballistic regimes prevail, initiating a power-law behavior of ∼ t2 . The GSER does, however, not include the presence of optical or external forces and considers that inertia is negligible. Therefore, the Mason-Weitz approximation is only valid for intermediate frequencies at negligible optical forces. In fact, Fig. 3(b) shows substantial deviations for G′′ (ω) data located out of an approximated frequency range between 3 × 103 and 3 × 105 Hz, where the MSD behaves as ∼ t. Nevertheless, the elastic component, G′ (ω), yields a good value for the spring constant with data outside that range. The elastic component for such a force is G′ (ω) = (6πa)/k. If G′ (103 Hz) = 0.33 Pa, as read from Fig.3(b), then we obtain k = 5.9 µm/N, which nicely agrees with the calculated value obtained from the calibration method and the NPAF linear regression. The viscosity value at a frequency of ω = 104 Hz is 0.99 mPa.s for water and 0.40 mPa.s for acetone, in good agreement with the formerly obtained values. The apparent simplicity of the Mason-Weitz methodology to estimate the moduli, G′ and G′′ , of complex fluids has made it the standard tool in microrheology for almost the last two decades. However, it has several known limitations. First, the GSER assumes that inertial effects are negligible and does not include external forces, such as the applied optical trap, which is commonly used in experimentsof high-frequency microrheology. Second, the unilateral Fourier transform of the MSD has to be evaluated from lag times t = 0 to ∞, which is not possible and, therefore, only limited frequency ranges can be reliably analyzed. Moreover, the GSER can break down for active and heterogeneous materials22 and, in that last case, a two-point MSD29 should be introduced in the GSER. For all these reasons, a microrheological analysis based alone on the GSER is not perfectly suitable to obtain viscoelastic moduli of Non-Newtonian fluids in the high-frequency range. A new methodology is hence required to overcome the limitations of the Mason-Weitz approach. One simple method is to remove the influence of the inertia of the solvent in the G′′ of a viscoelastic fluid by subtracting the solvent contribution by means of G′′ = G′′ (measured) − G′′ (buffer). A similar approach can be applied to the elastic component, which is very likely a superposition of the fluid elasticity and the stiffness of the optical trap. Evans et al.30 proposed to calculate G′ and G′′ directly in the time-domain, but they didn’t provide an expression for the relationship between the creep compliance J(t) and the MSD(t). Another promising work was published during 2012 by Indei, C´ordoba, Schieber and collaborators.31–34 These authors eliminate the influence of the bead and medium inertia as well as of the optical forces by proposing an Inertial Generalized Stokes Einstein Relationship (IGSER),

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(a)

m

0

), G''(

) (Pa),

-1

-2

3

Water, k = 6

m/N

m/N

10

1

e

G''(

)

1

10

0

10

G'(

)

-1

10

3

k=6 a G'(10 ) = 5.9

m/N

-2

10

Viscosity 4

m

2

G' (

ln NPAF

e

Acetone, k = 5

3

10

1

2

e

Mason-Weitz method d =1.88

(b)

(Pa.s)

e

Water, d = 1.88

= |G|/

-3

10

-3

0.0000

0.0002

0.0004

0.0006

3

10

4

5

10

10

(Hz)

Time t (s)

Figure 3. (a) Linear regressions (black lines) of the NPAF data (blue squares) calculated from the data of the MSD in water. The numbers 1 to 4 represent the same data as in Fig. 2(a). The spring constant values agree with the ones tabled in Fig 2(a). (b) Mechanical properties of water (blue) and acetone (red) at minimal optical trapping forces calculated using Mason-Weitz methodology.

assuming that the medium is incompressible. These methods are quite recent and have yet to be implemented and checked to be widely used for the analysis of high-frequency microrheological measurements.

ACKNOWLEDGMENTS P.D.G. acknowledges J.C. G´omez-S´ aez for her proofreading assistance and M.C.I under Project FIS2009-14008C02-02 for financial support. F.M.M. and S.J. acknowledge the National Center of Competence in Research “Nanoscale Science” and the Swiss National Science Foundation (SNF grant nos R’Equip 206021-121396 and 200021-143703)

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