Correlated fluctuations of microparticles in viscoelastic solutions: Quantitative measurement of material properties by microrheology in the presence of optical traps

The Brownian motions of microscopic particles in viscous or viscoelastic fluids can be used to measure rheological properties. This is the basis of recently developed one- and two-particle microrheology techniques. For increased temporal and spatial resolution, some microrheology techniques employ optical traps, which introduce additional forces on the particles. We have systematically studied the effect that confinement of particles by optical traps has on their auto- and cross-correlated fluctuations. We show that trapping causes anticorrelations in the motion of two particles at low frequencies. We demonstrate how these anticorrelations depend on trap strength and the shear modulus of viscoelastic media. We present a method to account for the effects of optical traps, which permits the quantitative measurement of viscoelastic properties in one- and two-particle microrheology over an extended frequency range in a variety of viscous and viscoelastic media.

Figure 1

(Color online) Schematic sketch of the two-particle microrheology experiment. A pair of silica particles (radius $R$) is trapped by a pair of laser traps at a separation distance $r$. The position fluctuations of each particle in the $x$ and $y$ directions are simultaneously detected with quadrant photodiodes. The cross-correlated position fluctuations are measured parallel and perpendicular to the line connecting the centers of the particles.

Figure 2

Displacement autocorrelations of a particle trapped in water as a function of frequency $f=\omega ∕2\pi$. The laser intensity (wavelength $\lambda =1064\mathrm{nm}$) was varied between 3 and $150\mathrm{mW}$. The corner frequency $f_c$ changes with laser intensity; the high frequency power-law slope is $-2$. Laser noise (increasingly prominent for stronger confinement at higher powers) was cut off at the low-frequency end of the curves.

Figure 3

Trap stiffnesses of two laser traps generated by independent lasers ($\lambda =1064$ and $830\mathrm{nm}$) measured from the fluctuations of two silica particles $(R=0.58\mu \mathrm{m})$ in water. The trap stiffnesses increased linearly with the laser intensities (measured before the beams entered the objective) for both traps.

Figure 4

Frequency dependence of (a) the real part ${\chi }_{\Vert }^{\prime \left(1\right)}$ and (b) the imaginary part ${\chi }_{\Vert }^{″\left(1\right)}$, of the measured single-particle response functions in water, for $x$ displacements (empty symbols). The data agree well with theoretical prediction (solid dark lines) calculated from Eq. (6a) without any adjustable parameters. In (b) the solid symbols are medium response functions ${\alpha }^{″\left(1\right)}$, calculated from ${\chi }_{\Vert }^{\prime \left(1\right)}$ and ${\chi }_{\Vert }^{″\left(1\right)}$ by eliminating the trap effect according to Eq. (6a). The solid gray line is the expected Stokes result for a sphere in a viscous fluid.

Figure 5

Apparent storage moduli (caused by the trap) as a function of frequency, measured in water with 1PMR for different laser powers (measured before the objective, see text). The solid lines are the theoretical predictions based on the measured trap stiffnesses.

Figure 6

Displacement autocorrelations as a function of frequency of a particle of radius $R=0.58\mu \mathrm{m}$ in a $10\mathrm{mg}∕\mathrm{ml}$ $fd$ virus solution for different laser intensities (measured before the objective, see legend). The sampling rate was $100\mathrm{kHz}$.

Figure 7

(a) Uncorrected storage moduli $G^{\prime }$ for a $10\mathrm{mg}∕\mathrm{ml}$ $fd$ solution measured with 1PMR using different laser powers (measured before the objective, see legend). The gray lines are the expected moduli corresponding to the $G^{\prime }$ measured in water for the same power settings. (b) Actual storage moduli $G^{\prime }$ calculated using the response functions corrected for the traps using Eq. (7a). (Data are shown versus frequency, $f=\omega ∕2\pi$).

Figure 8

(a) Uncorrected loss moduli $G^{″}$ and (b) actual loss moduli $G^{″}$ [obtained with Eq. (7a)], measured in $x$ direction with 1PMR for a $10\mathrm{mg}∕\mathrm{ml}$ $fd$ solution for different laser powers (measured before the objective, see legend). The gray line shows the loss modulus of the buffer. The trapping laser does not have a significant effect on $G^{″}$.

Figure 9

Inter-particle displacement correlation functions as a function of frequency, $f=\omega ∕2\pi$ of a pair of particles (radius $R=0.58\mu \mathrm{m}$) separated by a distance $r=2.9\mu \mathrm{m}$, trapped with different laser powers in water. Laser intensities were varied in parallel in the two traps (measured before the objective, legend: $1064∕830\mathrm{nm}$ powers). Low-frequency anticorrelations are not plotted in this ln-ln plot. At high frequencies, all curves superimpose onto one curve with slope $-2$.

Figure 10

(a) Real part and (b) imaginary part of the inter-particles response functions of two particles in water as a function of frequency, $f=\omega ∕2\pi$ in the parallel direction, trapped with different laser powers varied in parallel in the two traps (measured before the objective, legend: $1064∕830\mathrm{nm}$ powers). The theoretical predictions of Eq. (6c), calculated with the trap stiffness determined from corner frequency are plotted for comparison (lines).

Figure 11

(a) Apparent storage modulus $G^{\prime }$ and (b) apparent loss modulus $G^{″}$ (empty symbols) measured in parallel direction with 2PMR in water as a function of frequency, $f=\omega ∕2\pi$ with different laser powers varied in parallel in the two traps (measured before the objective, legend: $1064∕830\mathrm{nm}$ powers). The gray lines are the calculated apparent moduli $G^{\prime }$ and $G^{″}$ from the predicted response ${\chi }_{\Vert }$ as in Eq. (6c). Filled symbols in (b) show the corrected loss modulus of water calculated using ${\alpha }_{\Vert }$ and the expected loss modulus for water, $G^{″}\left(f\right)=-2\pi f\eta$ plotted as a broken line.

Figure 12

Inter-particle correlation functions in the parallel direction of two particles separated by a distance $r=2.9\mu \mathrm{m}$ in a $10\mathrm{mg}∕\mathrm{ml}$ $fd$ solution, as a function of frequency. The trap powers were varied in parallel (measured before the objective, legend: $1064∕830\mathrm{nm}$ powers). A power-dependent anticorrelation is observed at low frequencies.

Figure 13

Storage shear modulus of a $10\mathrm{mg}∕\mathrm{ml}$ $fd$ solution as a function of frequency $G^{\prime }$, measured with 2PMR in parallel direction. The trap powers were varied in parallel (measured before the objective, legend: $1064∕830\mathrm{nm}$ powers), with the (a) uncorrected modulus $G^{\prime }$ and (b) the actual modulus $G^{\prime }$ calculated using the corrected response functions in Eq. (7c).

Figure 14

Loss moduli of a $10\mathrm{mg}∕\mathrm{ml}$ $fd$ solution as a function of frequency $G^{″}$, measured with 2PMR in the parallel direction. The trap powers were varied in parallel (measured before the objective, legend: $1064∕830\mathrm{nm}$ powers). The uncorrected $G^{″}$ in (a) and actual modulus in (b) $G^{\prime }$ calculated using the corrected response functions [Eq. (7c)] do not show a significant difference for the lower powers. For higher laser powers $25∕50$, $70∕70$, and $200∕150\mathrm{mW}$ correction eliminates the negative values at low frequencies [not plotted in (a)].