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Keywords:

  • Brownian motion;
  • Einstein;
  • Langevin equation;
  • instantaneous velocity;
  • optical tweezers

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Brownian motion in a gas
  5. 3 Brownian motion in a liquid
  6. 4 Effects of detection noise on studying Brownian motion
  7. 5 Future
  8. Acknowledgements
  9. References
  10. Biographies

Brownian motion has played important roles in many different fields of science since its origin was first explained by Albert Einstein in 1905. Einstein's theory of Brownian motion, however, is only applicable at long time scales. At short time scales, Brownian motion of a suspended particle is not completely random, due to the inertia of the particle and the surrounding fluid. Moreover, the thermal force exerted on a particle suspended in a liquid is not a white noise, but is colored. Recent experimental developments in optical trapping and detection have made this new regime of Brownian motion accessible. This review summarizes related theories and recent experiments on Brownian motion at short time scales, with a focus on the measurement of the instantaneous velocity of a Brownian particle in a gas and the observation of the transition from ballistic to diffusive Brownian motion in a liquid.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Brownian motion in a gas
  5. 3 Brownian motion in a liquid
  6. 4 Effects of detection noise on studying Brownian motion
  7. 5 Future
  8. Acknowledgements
  9. References
  10. Biographies

Brownian motion is the apparently perpetual and random movement of particles suspended in a fluid (liquid or gas), which was first observed systematically by Robert Brown in 1827 [1]. When Brown used a simple microscope to study the action of particles from pollen immersed in water [1], he “observed many of them very evidently in motion”. The size of those particles was about 5 inline imagem. He also observed the same kind of motion with powders of many other materials, such as wood and nickel, suspended in water.

As first explained by Einstein in 1905 [2], the Brownian motion of a suspended particle is a consequence of the thermal motion of surrounding fluid molecules. Einstein's theory of Brownian motion predicts that

  • display math(1)

where inline image is the mean-square displacement (MSD) of a free Brownian particle in one dimension during time t , and D is the diffusion constant. The diffusion constant can be calculated by inline image, where inline image is the Boltzmann constant, T is the temperature, and inline image is the Stokes friction coefficient for a sphere with radius R. Here η is the viscosity of the fluid.

M. von Smoluchowski also derived the expression of MSD independently in 1906 [3], with a result that differed from Eq. (1) by a factor of about 2. In 1908, Paul Langevin introduced a stochastic force and derived Eq. (1) from Newton's second law [4, 5]. Langevin's approach is more intuitive than Einstein's approach, and the resulting “Langevin equation” has found broad applications in stochastic physics [6]. Experimental confirmation of Eq. (1) was provided by the brilliant experiments of Jean Perrin [7], recognized by the Nobel Prize in Physics in 1926. Theodor Svedberg also verified the Einstein-Smoluchowski theory of Brownian motion and won the Nobel Prize in Chemistry in 1926 for related work on colloidal systems [8].

Persistence and randomness are generally accepted as two key characteristics of Brownian motion. The trajectories of Brownian particles are classic examples of fractals [9]. They are commonly assumed to be continuous everywhere but not differentiable anywhere [10]. Since its trajectory is not differentiable, the velocity of a Brownian particle is undefined. According to Eq. (1), the mean velocity measured over an interval of time t is inline image. This diverges as t approaches 0, and therefore does not represent the real velocity of the particle [11, 12].

At short time scales (inline image, where inline image is the momentum relaxation time of a particle with mass M), the dynamics of a Brownian particle is expected to be dominated by its inertia and its trajectory cannot be self-similar. This is termed “ballistic Brownian motion” to be distinguished from the common “diffusive Brownian motion”. Figure 1 shows a 2D trajectory of a Brownian particle. The black curve is assumed to be a true trajectory of the particle. Red dots are measured positions. In Fig. 1 A, the sampling rate is too small to measure the velocity of the Brownian particle. The measured trajectory (red curve) is completely different from the true trajectory, and appears chaotic. It is impossible to obtain the velocity of the particle from the measured trajectory in Fig. 1 A. In Fig. 1 B, the sampling rate is much larger. Now the measured trajectory is very close to the true trajectory of the particle. If the measured displacement of the particle is inline image during time inline image, then the velocity of the particle is approximately inline image.

image

Figure 1. (online color at: www.ann-phys.org) A 2D trajectory of a Brownian particle. The black curve is assumed to be a true trajectory of the particle. Red dots are measured positions, and red curves are measured trajectories. The sampling rate of (B) is 10 times of that of (A).

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In 1900, F. M. Exner made the first quantitative study of Brownian motion by measuring the velocity of Brownian particles suspended in water [13, 14]. He found that the measured velocity decreased with increasing particle size and increased with increasing water temperature. However, his measured velocities were almost 1000-fold smaller than those predicted by the energy equipartition theorem [13]. The reason of this discrepancy was not understood until A. Einstein developed his kinetic theory about Brownian motion [2].

In 1907, Einstein published a paper entitled “Theoretical observations on the Brownian motion” in which he considered the instantaneous velocity of a Brownian particle [11, 12]. Einstein showed that by measuring this quantity, one could prove that “the kinetic energy of the motion of the centre of gravity of a particle is independent of the size and nature of the particle and independent of the nature of its environment”. This is one of the basic tenets of statistical mechanics, known as the equipartition theorem. However, Einstein concluded that due to the very rapid randomization of the motion, the instantaneous velocity of a Brownian particle would be impossible to measure in practice [11, 12]:

“We must conclude that the velocity and direction of motion of the particle will be already very greatly altered in the extraordinary short time θ, and, indeed, in a totally irregular manner. It is therefore impossible – at least for ultramicroscopic particles – to ascertain inline image by observation.”

Einstein's conclusion was unchallenged for more than 100 years because of the exclusive difficulty of such a measurement. For a 1 inline imagem diameter silica (SiO2) sphere in water at room temperature, the momentum relaxation time inline image is about 0.1 inline images and the root mean square (rms) velocity inline image is about 2 mm/s in one dimension. To measure the instantaneous velocity with 10% uncertainty, one would require 2 pm spatial resolution and 10 ns temporal resolution, which is a very difficult task. Due to the lower viscosity of gas as compared to liquid, the momentum relaxation time inline image of a particle in air is much larger. This lowers the requirements of both temporal and spatial resolution.

Nondiffusive Brownian motion of colloidal suspensions with high concentrations at short time scales have been studied by measuring the autocorrelation functions of multiply scattered, transmitted light [15-17]. Recent developments in optical tweezers and detection systems with unprecedented resolution now prove to be an indispensable tool for studying the Brownian motion of a single particle at short time scales [18-26]. For example, we have observed the ballistic Brownian motion and measured the instantaneous velocity of a Brownian particle for the first time with an optically trapped bead in air [21]. Huang et al. have observed the transition from ballistic Brownian motion to diffusive Brownian motion in a liquid [24]. Franosch et al. have observed resonances arising from hydrodynamic memory in Brownian motion in a liquid and the long-sought colored spectrum of the thermal force [25, 27].

Besides Brownian motion at short time scales, theoretical and experimental studies of anisotropic Brownian motion and Brownian motion in nonequilibrium systems are currently pursued by many groups. For example, several groups reported anisotropic Brownian motion of particles near interfaces [28-32], and Brownian motion of anisotropic particles such as ellipsoids [33, 34], nanotubes [35, 36] and helical bacteria [37]. Brownian motion in nonequilibrium systems is of particular interest because it is directly related to the transport of molecules and cells in biological systems. Important examples include Brownian motors [38, 39], active Brownian motion of self-propelled particles [40-46], hot Brownian motion [47], and Brownian motion in shear flows [48]. Recent theoretical studies also found that the inertias of particles and surrounding fluids can significantly affect the Brownian motion in nonequilibrium systems [49-54].

In this review, Section 'Brownian motion in a gas' introduces the theories of Brownian motion of particles in a gas, and the recent measurement of the instantaneous velocity of a Brownian particle in air. Section 'Brownian motion in a liquid' introduces the theories of Brownian motion of particles in a liquid at short time scales, the experimental observation of the colored thermal force, and the transition from ballistic to diffusive Brownian motion in a liquid. Section 'Effects of detection noise on studying Brownian motion' discusses the effects of detection noise on the measurement of different quantities of Brownian motion. Finally, in Section 'Future', we discuss future experiments on Brownian motion at short time scales.

2 Brownian motion in a gas

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Brownian motion in a gas
  5. 3 Brownian motion in a liquid
  6. 4 Effects of detection noise on studying Brownian motion
  7. 5 Future
  8. Acknowledgements
  9. References
  10. Biographies

In this section, we assume that the density of a gas is much smaller than the density of the suspended Brownian particles. So the inertia effects of the gas can be neglected.

2.1 Theory

The mean free path of molecules in air at 1 atmosphere at room temperature is about 68 nm [55]. The collision rate between a 1-inline imagem-diameter microsphere suspended in air and surrounding air molecules is about 1016 Hz at ambient conditions. The observed Brownian motion is an averaged effect of these ultrafast collisions. Because of the huge difference between the mass of a microsphere and that of an air molecule, the motion of a microsphere can only be changed significantly by a large number of collisions. This is reflected in the fact that inline images for a 1-inline imagem-diameter silica microsphere in air at ambient conditions. The dynamics of a Brownian particle at time scales much longer than that of individual collisions can be described by a Langevin equation [4, 5]. Here we introduce the theory of Brownian motion in a gas following the classic work of Uhlenbeck and Ornstein [56].

2.1.1 A free particle in a gas

The dynamics of a Brownian particle with mass M in a gas can be described by a Langevin equation [4, 5, 56]:

  • display math(2)

where

  • display math(3)

is the Brownian stochastic force. inline image is a normalized white-noise process. Hence for all t and inline image,

  • display math(4)

The MSD for a Brownian particle at thermal equilibrium with the air is [56]:

  • display math(5)

where inline image is the damping coefficient. We have inline image. At long time scales (inline image), the MSD is the same as that predicted by Einstein's theory (Eq. (1)). At very short time scales, the MSD is

  • display math(6)

The velocity autocorrelation function is [56]:

  • display math(7)

Although these equations are initially derived for an ensemble of particles, the ergodic theorem dictates that they are also valid for measurements of a single particle taken over a long time.

The damping coefficient Γ0 in a gas can be calculated by kinetic theory. Assuming the reflection of gas molecules from the surface of a microsphere is diffusive, and the molecules thermalize with the surface during collisions, we obtain [57, 58]

  • display math(8)

where η is the viscosity coefficient of the gas, R is the radius of the microsphere, and inline image is the Knudsen number. Here s is the mean free path of the gas molecules. inline image is a small positive function of inline image [58]. At high pressures where inline image, the damping coefficient is inline image, which is the same as the prediction of Stokes’ law.

At very short time scales, the motion is ballistic and its instantaneous velocity can be measured as inline image, when inline image [56]. The ballistic Brownian motion is different from a simple ballistic motion. For a simple ballistic motion with velocity u, we have inline image and inline image. The velocity u can be any value and usually has no relation with the temperature of the environment. For the ballistic Brownian motion, the amplitude of the velocity is determined by the temperature of the environment. The 1D Maxwell-Boltzmann distribution of the velocity of a particle in thermal equilibrium is

  • display math(9)

where inline image is the velocity of the particle along direction i, which can be any direction.

2.1.2 An optically trapped microsphere in a gas

For small displacements, the effect of optical tweezers on the microsphere's motion can be approximated by that of a harmonic potential. The Brownian motion of a particle in a harmonic trap has been studied by Uhlenbeck and Ornstein [56], and Wang and Uhlenbeck [59]. The equation of the Brownian motion of a microsphere in a harmonic trap is:

  • display math(10)

where inline image is the natural angular frequency of the trapped microsphere when there is no damping, and inline image. The cyclic frequency of the damped oscillator is inline image. The system is underdamped when ω1 is real (inline image), critically damped when inline image, and overdamped when ω1 is imaginary (inline image).

The MSD of a Brownian particle in an underdamped harmonic trap in air is [59]:

  • display math(11)

The position autocorrelation function is related to the MSD by:

  • display math(12)

where inline image. The rms amplitude is inline image. The normalized position autocorrelation function (PACF) of the particle is [59]:

  • display math(13)

The normalized velocity autocorrelation function (VACF) of the particle is [59]:

  • display math(14)

Both the position autocorrelation function and the velocity autocorrelation function oscillate for an underdamped system.

Similar to the optical spectrum of an atom, the power spectrum of the Brownian motion of a trapped microsphere contains a lot of information about the system. The power spectral density (PSD) of a variable is the squared modulus of its Fourier transform [59-61]. The expected values of the PSD of an optically trapped microsphere in air is

  • display math(15)

Eq. (15) is valid for both underdamped [57] and overdamped systems. The measured PSD of a recorded inline image is inline image, which depends on a sample of the white noise inline image. Thus an experimental PSD will appear noisy. Averaging many measured inline image will result in a spectrum close to the expected spectrum inline image. Another way to reduce the noise in a measured spectrum is “blocking”[60]. A “block” of consecutive data points (inline image,inline image,inline image) can be represented by a single new “data point” (inline image,inline image) which are the block averages.

2.2 Experimental observation of the instantaneous velocity of a Brownian particle in air

Because of the lower viscosity of air as compared to that of liquid, a particle suspended in air is an ideal system for studying the ballistic Brownian motion. The main difficulty of studying Brownian motion in air, however, is that the particle will fall under the influence of gravity. To overcome this problem, Fedele et al. used aerosol suspensions with small particles (inline imagem diameter) to achieve a long sedimentation time [62], Blum et al. performed the experiment under microgravity conditions in the Bremen drop tower [63]. Due to the lack of a detection system with a sufficient resolution, neither experiment was able to measure the instantaneous velocity of Brownian motion. We overcame this problem by using optical tweezers to simultaneously trap and monitor a silica bead in air and vacuum, allowing long-duration, ultra-high-resolution measurements of its motion. Here we review this experiment that was originally reported in Ref. [21] with more experimental details.

2.2.1 A fast detection system

In order to measure the instantaneous velocity of the Brownian motion of a trapped microsphere in air, we implemented an ultrahigh resolution detection system.

We used an ultra-stable NPRO laser (Model: 126-1063-700, Lightwave Electronics (now JDSU)) to trap and monitor a silica bead in vacuum. Its rms intensity noise is < 0.05 % over the range from 10 Hz to 2 MHz, and is shot noise limited above 10 MHz. It is a single frequency laser with a wavelength of 1064 nm and a coherence length longer than 1000 m. A detailed characterization of this type of laser can be found in Ref. [64]. We used this laser for both trapping and detection. This is achieved by using a polarizing beam splitter cube to reflect one of the trapping beams for detection (Fig. 2).

image

Figure 2. (online color at: www.ann-phys.org) This simplified schematic shows our counter-propagating dual-beam optical tweezers, and a novel detection system (Figure adapted from Ref. [21]). The s-polarized beam is reflected by a polarizing beam splitter cube after it passes through a trapped bead inside a vacuum chamber. Then, for detection, it is split by a mirror with a sharp edge. The p-polarized beam passes through the cube.

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Our lab has previously developed a fast position-sensitive laser beam detector [20]. The previous detector used a bundle of optical fibers that spatially splits the incident beam, and a fast balanced photodetector to measure the difference between the two halves of the beam. We simplified the detection system by using a mirror with a sharp edge (BBD05-E03, Thorlabs) to replace the fiber-optic bundle for splitting the beam (Fig. 2). The sharp edge of a mirror is much smoother than the boundary between the two halves of a fiber bundle. So it is much simpler and has less noise than a fiber bundle for splitting the laser beam.

We used a balanced detector (PDB120C, photodiode diameter: 0.3 mm, Thorlabs) with a bandwidth of 75 MHz for detection. The detector is sensitive to light with wavelengths in the range of 800–1700 nm. It has a high transimpedance gain of 1.8 × 105 V/A. The detector measures the difference between the two halves of the beam, which is proportional to the particle excursion. The intensity noise of the laser is contained in both halves and is thus canceled in the measurement. This detection system enables us to monitor the real-time position of a trapped microsphere in air with Ångstrom spatial resolution and microsecond temporal resolution [21]. We have also developed a 3D detection system that can monitor the 3D motion of a microsphere trapped in vacuum with a sensitivity of about 39 fm Hzinline image over a wide frequency range [57].

2.2.2 Experimental results

A simplified scheme of our setup for measuring the instantaneous velocity of a Brownian particle in air is shown in Fig. 2. The trap is formed inside a vacuum chamber by two counter-propagating laser beams focused to the same point by two identical aspheric lenses with focal length of 3.1 mm. The two 1064 nm laser beams are orthogonally polarized, and their frequencies differ by 160 MHz to avoid interference.

The two laser beams are aligned with the help of a pinhole aperture whose diameter is 1.0 ± 0.5 inline imagem. We intentionally make the waist of one beam larger than the other to make this alignment less critical. The measured waists of the two beams are about 2.2 inline imagem and 3.0 inline imagem, respectively. Once a bead is trapped, we keep the power of one beam constant, and tune the power of the other beam to maximize the trapping frequency.

When the bead deviates from the center of the trap, it deflects both trapping beams. We monitor the position of the bead by measuring the deflection of one of the beams, which is split by a mirror with a sharp edge. This simple, yet novel, detection scheme has a bandwidth of 75 MHz and ultra-low noise [65, 20]. The position signal of a trapped bead is recorded at a sampling rate of 2 MHz. Because of the detection noise, we are not able to obtain accurate instantaneous velocities of a bead at this rate. To reduce the noise, we average every 10 successive position measurements, and use these averages to calculate instantaneous velocities with time resolution of 5 inline images. Although this method reduces the temporal resolution by a factor of 10, it greatly increases the signal-to-noise ratio if both the trapping period (inline image) and momentum relaxation time are much larger than 5 inline images. These conditions are satisfied here since the trapping period is about 320 inline images, inline image inline images at 99.8 kPa (749 torr), and inline image inline images at 2.75 kPa (20.6 torr).

Figure 3 shows typical samples of position and velocity traces of a trapped bead. The position traces of the bead at these two pressures appear very similar. The instantaneous velocity of the bead at 99.8 kPa changes more frequently than that at 2.75 kPa, because the momentum relaxation time is shorter at higher pressure.

image

Figure 3. (online color at: www.ann-phys.org) One-dimensional trajectories of a 3 inline imagem diameter silica bead trapped in air at 99.8 kPa (A) and at 2.75 kPa (B). The instantaneous velocities of the bead corresponding to these trajectories are shown in (C) and (D). Figure adapted from Ref. [21].

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Figure 4 shows the mean square displacements of a 3 inline imagem silica bead as a function of time. The measured MSD's fit excellently with Eq. (11) over three decades of time for both pressures. The measured MSD's are completely different from those predicted by Einstein's theory of Brownian motion in a diffusive regime. The slopes of measured MSD curves at short time scales are double of those of the MSD curves of diffusive Brownian motion in the log-log plot (Fig. 4). This is because the MSD is proportional to t2 for ballistic Brownian motion, and it is proportional to t for diffusive Brownian motion. Another important feature is that the MSD curves are independent of air pressure at short time scales, as is predicted by Eq. (6) for ballistic Brownian motion, whereas the MSD in the diffusive regime does depend on the air pressure. At long time scales, the MSD oscillates and saturates at a constant value because of the optical trap.

image

Figure 4. (online color at: www.ann-phys.org) The mean square displacements of a 3 inline imagem silica bead trapped in air at 99.8 kPa (red square) and 2.75 kPa (black circle). The solid lines are the theoretical predictions of Eq. (11). The prediction of Einstein's theory of free Brownian motion in the diffusive regime is shown in dashed lines for comparison. Figure adapted from Ref. [21].

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The distributions of the measured instantaneous velocities are displayed in Fig. 5. They agree very well with the Maxwell-Boltzmann distribution. The measured rms velocities are inline image = 0.422 mm/s at 99.8 kPa and inline image = 0.425 mm/s at 2.75 kPa. These are very close to the prediction of the energy equipartition theorem, inline image, which is 0.429 mm/s. As expected, the velocity distribution is independent of pressure. Thus the Maxwell-Boltzmann distribution of velocities and the equipartition theorem of energy for Brownian motion were verified directly.

image

Figure 5. (online color at: www.ann-phys.org) The distribution of the measured instantaneous velocities of a 3 inline imagem silica bead. The statistics at each pressure are calculated from 4 million instantaneous velocities. The solid lines are Maxwell-Boltzmann distributions. Figure adapted from Ref. [21].

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Figure 6 shows the normalized VACF of the bead at two different pressures. They fit with Eq. (14) nicely. At 2.75 kPa, one can clearly see the oscillations due to the optical trap. Eq. (14) is independent of the calibration factor of the detection system. The only independent variable is time t, which we can measure with high precision. Thus the normalized VACF provides an accurate method to measure inline image and ω0. We can also calculate the diameter of the silica bead from the inline image value at 99.8 kPa [66]. The obtained diameter for this microsphere is 2.79 inline imagem. This is within the uncertainty range given by the supplier of 3.0 inline imagem silica beads. We use this value in the calculation of MSD and normalized VACF.

image

Figure 6. (online color at: www.ann-phys.org) The normalized velocity autocorrelation functions of the 3 inline imagem bead at 99.8 kPa (red square) and at 2.75 kPa (black circle) from the measurements. The solid lines are fittings with Eq. (14). Figure adapted from Ref. [21].

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For a particle at a certain pressure and temperature, inline image should be independent of the trapping frequency. We verified this by changing the total power of the two laser beams from 25 mW to 220 mW. The measured inline image of a microsphere trapped at 19.6 torr and 749 torr as a function of the total laser power is shown in Fig. 7. Here we use a new microsphere and a smaller data set to calculate the inline image than those used in the previous figures. Although the data points for each pressure are not perfectly on a line, it is clear that the inline image’s are independent of the laser power within the experimental uncertainty. Fitting the data for each pressure with a straight line, we obtain inline image at 2.61 kPa, and inline image at 99.8 kPa for this microsphere, where P is the total power of the two trapping beams. Thus inline image changes less than 1.3% for both pressures when the total laser power is changed from 0 to 230 mW. This proves that the fitting method is very accurate, and the heating due to the laser beams (which would change the viscosity [67, 68] and affect inline image) is negligible at these pressures.

image

Figure 7. (online color at: www.ann-phys.org) Measured momentum relaxation times (inline image) of a microsphere trapped at 2.61 kPa and 99.8 kPa as a function of the total power of the two trapping beams.

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3 Brownian motion in a liquid

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Brownian motion in a gas
  5. 3 Brownian motion in a liquid
  6. 4 Effects of detection noise on studying Brownian motion
  7. 5 Future
  8. Acknowledgements
  9. References
  10. Biographies

The main difference between the Brownain motion in a liquid and that in a gas is the hydrodynamic effects of the liquid [46]. The Brownian motion of colloidal particles in a liquid at high concentrations have been studied with diffusing wave spectroscopy, which requires each photon to be scattered many times before reaching the detector [15-17]. The recent developments in optical tweezers provide a new tool for studying the Brownian motion of single particles with unprecedented precision [18, 20, 21, 24, 25]. Meanwhile, precise calibrations (force, position, etc.) of optical tweezers demand better understanding of the Brownian motion of trapped particles [60, 69, 68, 70].

Table 1. Characteristic time scales of an optically trapped silica microsphere in water at 20 °C. Some examples of the spring constant of the optical trap (k) are shown in the 5th column. It is assumed to be inversely proportional to the diameter of the microsphere when the laser power is constant
Diameterinline imageinline imageinline imagekinline image
(inline imagem)(inline images)(inline images)(ns)(inline imageN/m)(inline images)
1.00.110.250.3410094
3.01.02.21.0133.3851
4.72.455.511.5821.32083
1011.125.03.4109443

3.1 Theory

Besides the inertia of the particle itself, the inertia of the surrounding liquid is also important for Brownian motion of particles in a liquid. The motion of a particle will cause long-lived vortices in the liquid that will affect the motion of the particle itself. This is the hydrodynamic memory effect of the liquid, which dominates the dynamics of the particle at short time scales. These hydrodynamic memory effects were first studied by Vladimirsky in 1945 [71]. In 1960s, several authors found in computer simulations that the velocity autocorrelation function (VACF) of fluid molecules had a power-law tail in the form of inline image [72, 73], in contrast to the exponential decay in a dilute gas. Hinch obtained an analytical solution of the VACF for free particles from the original Langevin analysis [74]. Clercx and Schram calculated the MSD and VACF of a Brownian particle in a harmonic potential in an incompressible liquid [75], which can be used to describe the Brownian motion of an optically trapped microsphere in a liquid directly [30, 24, 76].

3.1.1 A free particle in a liquid

The effective mass of the microsphere in an incompressible liquid is the sum of the mass of the microsphere and half of the mass of the displaced liquid [77, 78]:

  • display math(16)

where inline image is the mass of the microsphere, inline image is the mass of displaced liquid, inline image is the density of the microsphere, and inline image is the density of liquid. The energy equipartition theorem needs to be modified to:

  • display math(17)

where v is the velocity of the microsphere in one dimension. Thus the rms velocity is inline image. Because of the memory effect of liquid, the velocity autocorrelation function (VACF) of a free particle in a liquid will not be inline image as in air, but [74, 46, 75, 79]

  • display math(18)

where

  • display math(19)

inline image is the momentum relaxation time of the particle due to its own inertia, inline image characterizes the effect of liquid. Here η is the viscosity of liquid and R is the radius of the microsphere.

At long time scales, Eq. (18) approaches

  • display math(20)

At short time scales, Eq. (18) approaches

  • display math(21)

where

  • display math

For a silica microsphere in water, inline image. The normalized VACF approaches 1 at short time scales as

inline image, rather than inline image. Thus the dynamics of the particle is dominated by the hydrodynamic effects of the liquid. This is very different from the case in air.

3.1.2 An optically trapped microsphere in a liquid

The optical trap provides a harmonic force inline image on the microsphere when the displacement of the microsphere is small. inline image where Ω is the natural angular frequency of the trap. Clercx and Schram [75] gave analytical solutions for the MSD and VACF of a trapped Brownian particle in a liquid, and Berg-Sørensen and Flyvbjerg [60] gave a solution for the power spectrum density (PSD) of a trapped Brownian particle in a liquid. This section introduces their analytical solutions and provides some numerical results to visualize those solutions.

Because the velocity of the Brownian motion of a microsphere in liquid is much smaller than the speed of sound in the liquid, the fluid motion can be described by the linearized incompressible time-dependent Navier-Stokes equation. The Langevin equation of the motion of a trapped microsphere in an incompressible liquid is [75]:

  • display math(22)

The first term after the equal sign of Eq. (22) is the harmonic force, the second term is the ordinary Stokes's friction, the third term is a memory term associated with the hydrodynamic retardation effects of the liquid, and the last term is the Brownian stochastic force.

By the famous fluctuation-dissipation theorem, the thermal force is directly related to the frictional force. So the hydrodynamic memory of the liquid will affect both the thermal force and the frictional force. The thermal force is not a white noise, but becomes colored. The correlation in the thermal force is [25, 46, 75, 27]

  • display math(23)

which is very different from a delta function (Eq. (4)).

The mean-square displacement of a trapped microsphere in a liquid is [75, 80]

  • display math(24)

The coefficients z1, z2, z3, and z4 are the four roots of the equation [80]

  • display math(25)

where inline image. For inline image, Eq. (24) approaches

  • display math

The normalized VACF of a trapped microsphere in a liquid is [75, 80]

  • display math(26)

The power spectral density is [60, 80]:

  • display math(27)

where f is the observation frequency, inline image is the corner frequency of the power spectrum due to the trap, and inline image. For inline image, Eq. (27) approaches

  • display math

where inline image.

At inline image, Eq. (26) predicts inline image = inline image, which is different from the energy equipartition theorem inline image = inline image. This conflict is caused by the assumption in Eq. (22) that the liquid is incompressible. For inline image, we need to consider the liquid to be compressible. Here inline image is the time required for a sound wave to travel a sphere radius, where c is the speed of sound in the liquid. The effects of compressibility have been studied by Zwanzig and Bixon [77]. The normalized velocity autocorrelation function at inline image is [77]:

  • display math(28)

where

  • display math(29)
  • display math(30)

At very short time scales inline image, Eq. (28) approaches inline image. The short time limit inline image because the normalization factor is inline image in Eq. (28), rather than inline image.

The calculated MSD's of microspheres with different diameters in water are shown in Fig. 8, and the corresponding power spectra are shown in Fig. 9. Fig. 10 displays the normalized velocity autocorrelation function (inline image) of an optically trapped silica microsphere in water at 20 °C. The thick solid lines are calculated from Eq. (26), which treats the water as an incompressible fluid. The thin solid lines at short time scales (inline image s) are calculated from Eq. (28), which includes the compressibility effects of water. The dashed lines are exponential decays with inline images, corresponding to a 3.0 inline imagem microsphere. As clearly shown in Fig. 10, the VACF of a microsphere in water is very different from exponential decay because of the hydrodynamic memory effects of water.

image

Figure 8. (online color at: www.ann-phys.org) Calculated mean square displacement of an optically trapped silica microsphere in water at 20 °C. Parameters are the same as those in Table 1.

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image

Figure 9. (online color at: www.ann-phys.org) Calculated power spectra of an optically trapped silica microsphere in water at 20 °C. Parameters are the same as those in Table 1.

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image

Figure 10. (online color at: www.ann-phys.org) Calculated normalized velocity autocorrelation function of an optically trapped silica microsphere in water at 20 °C. Parameters are the same as those in Table 1. The thin solid lines (inline image s) are calculated from Eq. (28), and the thick solid lines are calculated from Eq. (26). The dashed lines are exponential decays with inline images, corresponding to a 3.0 inline imagem microsphere.

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The thin solid lines are expected to be correct for inline image, and the thick sold lines are expected to be correct for inline image. The intermediate regime inline image is poorly understood. A recent experiment has measured the VACF of a Brownian particle in water at VACF<0.35 [24, 30]. A measurement of the VACF between 1 and 0.35 is required in order to better understand the hydrodynamic effects and compressibility effects of water on Brownian motion [81-84].

3.2 Experimental observation of the color of thermal force in a liquid

Conventionally, the thermal force exerted on a Brownian particle are assumed to be a white noise. Due to the hydrodynamic memory effects of the liquid, this thermal force is in fact colored. Direct experimental observation of the color of the thermal force in a liquid was elusive until the recent work by Franosch et al. [25]. They used optical tweezers with very large stiffness (inline imageN/m for 2.9 inline imagem diameter particles) to study the Brownian motion of a trapped microsphere. Their key insight is that at long times, strong trapping eventually dominates over friction and becomes the main force counteracting thermal force [27]. So the Langevin equation reduces to inline image. Consequently, the correlations in the thermal force can be obtained by the position autocorrelation function inline image.

Figure 11 shows the results of Franosch et al.’s experiment on the color of thermal force [25]. The data in Fig. 11(a) clearly shows the departure of thermal force from a white noise whose power spectrum is a horizontal line. The measured spectrum of the thermal force increases at higher frequencies. Franosch et al. also observed resonances in Brownian motion in liquid where overdamped motion is often assumed. In order to observe the resonance, they used optical tweezers with very large stiffness, and acetone as a liquid rather than water. The viscosity of acetone is about 3 times smaller than that of water. So the motion of a trapped bead is less damped in acetone than in water. The resonance can be clearly seen in Fig. 11(b). Remarkably, this resonance is mainly due to the inertia of the liquid, rather than the inertia of the particle itself.

image

Figure 11. (online color at: www.ann-phys.org) The color of thermal force (Figure adapted from Ref. [25]). (a), The normalized power spectral density (PSD) of thermal force of an optically trapped melamine resin sphere (R = 1.45 inline imagem) in water (green circles) or acetone (blue circles). The black lines are predictions of the hydrodynamic theory. (b), The normalized power spectral density of position.

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3.3 Experimental observation of the transition from ballistic to diffusive Brownian motion in a liquid

Recently, Huang et al. studied the Brownian motion of a single particle in an optical trap in water with sub-Angstrom resolution and measured the velocity autocorrelation function of the Brownian motion [24]. In their experiments, inline image due to the confinement of the optical tweezers was typically two orders of magnitude larger than inline image. So the role of the optical confinement can be neglected during the transition from ballistic to diffusive Brownian motion.

Figure 12 shows Huang et al.’s results of MSD's for 1 inline imagem and 2.5 inline imagem silica particles from 10 ns to 1 ms [24]. The MSD's increase at short time scales, and reach plateaus at long time scales due to the confinement of the optical trap. At short time scales, the particle undergoes free, but correlated Brownian motion because of the inertia of the particle and the surrounding liquid. The effect of the particles’ inertia on Brownian motion is visible in Fig. 12. The slopes of the MSDs are close to 2 at short time scales, as predicted for ballistic Brownian motion. It is remarkable that Huang et al. achieved a temporal resolution of about 10 ns in the case of a 1 inline imagem silica particle. At this temporal resolution, Huang et al. resolved a MSD as small as 0.0005 nm2, corresponding to 22 pm spatial resolution. The temporal resolution shown in Fig. 12 is about 0.1 inline image for a 1 inline imagem particle, and about 0.15 inline image for a 2.5 inline imagem particle.

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Figure 12. (online color at: www.ann-phys.org) Measured mean square displacement of silica spheres with diameters of 1 inline imagem and 2.5 inline imagem (Figure adapted from Ref. [24]). The red lines with slope 2 show the expected behavior of ballistic Brownian motion.

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Figure 13 shows Huang et al.’s experimental results of the VACF of a 2 inline imagem resin particle and a theoretical fit with the Clercx-and-Schram theory (Eq. (26)) [75]. The predictions of VACF from Langevin equations neglecting the hydrodynamic effects of water are also displayed for comparison. The experimental results agree with the Clercx-and-Schram theory. For times shorter than inline images, the velocity correlations are smaller than the predictions of the Langevin equation neglecting the inertia of the fluid. At longer times, the correlations are stronger because of the vortex developed in the fluid. The small anti-correlation dip near 300 inline images is because of the optical trap.

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Figure 13. (online color at: www.ann-phys.org) Normalized experimental VACF for a 2 inline imagem diameter resin sphere in an optical trap (Figure adapted from Ref. [24]). The blue line shows the fitting of the Clercx-and-Schram theory (Eq. (26)) to the whole experimental VACF. The grey (dashed) and green lines show the expectations of VACF for the same particle if the inertia of the liquid is ignored, in the absence (grey) and presence (green) of the harmonic trap.

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Although the spatial resolution of Huang et al.’s experiment is sufficient to observe the transition from ballistic to diffusive Brownian motion in MSD and even compute a VACF, it is not enough to measure the instantaneous velocity of Brownian motion in a liquid. The reason is that both MSD and VACF are insensitive to white noise of the detection system since they are averages over large data sets. On the other hand, the measurement of the instantaneous velocity is very sensitive to the detection noise. We will discuss about this issue in the following section.

4 Effects of detection noise on studying Brownian motion

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Brownian motion in a gas
  5. 3 Brownian motion in a liquid
  6. 4 Effects of detection noise on studying Brownian motion
  7. 5 Future
  8. Acknowledgements
  9. References
  10. Biographies

In the presence of detection noise [20, 21, 24], the measured position of the microsphere can be expressed as

  • display math(31)

where inline image is the real position of the microsphere, and inline image is the noise of the detection system. The mean square displacement (MSD) of the measured positions is [79]:

  • display math(32)

where the average is taken over all possible t0. This derivation assumes no correlation between the real position of the microsphere and the detection noise. In this case, the real MSD of the microsphere can be obtained by inline image, as in Ref. [18, 24]. inline image is usually independent of time, as shown in Fig. 4.

The measured velocity of the microsphere is

  • display math(33)

where inline image. Because the measured velocity contains a noise signal inline image, the smallest inline image of the detection system may not be the best value for measuring the velocity. The data acquisition (DAQ) card creates noise when it converts an analog signal to a digital signal due to the finite number of bits. The minimum value of inline image is limited by the DAQ card, thus inline image may be larger than the real velocity of the microsphere (inline image) if inline image is too small.

The measured velocity represents the real instantaneous velocity of the microsphere if inline image and inline image is negligible. This requires inline image. One can check whether this condition is satisfied by comparing the signal when a microsphere is trapped in the optical tweezer and when there is no microsphere in the optical tweezer. The relation between inline image and inline image can be obtained from the measured inline image’s:

  • display math(34)

Thus inline image is equivalent to inline image.

The measured velocity autocorrelation function is

  • display math(35)

If the noise of the detection system has almost no correlation (white noise), the last term of this equation can be neglected. Thus

  • display math(36)

So the measurement of the velocity autocorrelation function is not sensitive to the noise of the detection system [24]. On the other hand, the measurement of the instantaneous velocity is very sensitive to the noise of the detection system [21].

If the detection system samples the position of the microsphere every inline image that is much shorter than the required temporal resolution inline image, we can reduce the noise in the measured velocity by using successively averaged positions to calculate the velocity. Let inline image, where inline image, then

  • display math(37)

The measured velocity becomes

  • display math(38)

Then the velocity noise is

  • display math(39)

On average, the rms amplitude of inline image in Eq. (39) is inline image times smaller than that of inline image if the position noise inline image is a white noise. If inline image, the noise can be reduced by a factor of 32, which is a significant number. The condition of using the averaging method is that inline image. So the detection system must have a very high temporal resolution. This averaging method has been utilized in both the measurement of the VACF of a Brownian particle in a liquid [24] and the measurement of the instantaneous velocity of a Brownian particle in air [21].

5 Future

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Brownian motion in a gas
  5. 3 Brownian motion in a liquid
  6. 4 Effects of detection noise on studying Brownian motion
  7. 5 Future
  8. Acknowledgements
  9. References
  10. Biographies

As shown in this review, optical tweezers have become an indispensable tool for studying Brownian motion at short time scales. The instantaneous velocity of a Brownian particle trapped in a gas has been measured [21]. Recently, the velocity autocorrelation function of a Brownian particle in water was measured successfully for inline image [30, 24, 79].

The instantaneous velocity of a Brownian particle in liquid is much more difficult to measure, and has not been measured to date. A successful measurement of the instantaneous velocity of a Brownian particle in a liquid will complete the task that was considered by Einstein more than 100 years ago [11, 12] and open a new door for studying Brownian motion. For example, the results can be used to test the modified energy equipartition theorem inline image = inline image and the Maxwell-Boltzmann velocity distribution. A measurement of the VACF > 0.35 will deepen our understanding of the hydrodynamic effects and compressibility effects of a liquid on Brownian motion [81-84]. The ability to measure the instantaneous velocity of a Brownian particle will be invaluable in studying nonequilibrium statistical mechanics [85, 86]. The Brownian motion of a suspended particle can be used for microrheology to probe the properties of fluids, such as viscoelastic fluids [87-89], and surrounding environments [90-92].

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Brownian motion in a gas
  5. 3 Brownian motion in a liquid
  6. 4 Effects of detection noise on studying Brownian motion
  7. 5 Future
  8. Acknowledgements
  9. References
  10. Biographies

M.G.R. acknowledges support from the Sid W. Richardson Foundation and the R. A. Welch Foundation grant number F-1258. We would like to thank S. Kheifets and D. Medellin for their help in the original experiment which is an important part of this review.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Brownian motion in a gas
  5. 3 Brownian motion in a liquid
  6. 4 Effects of detection noise on studying Brownian motion
  7. 5 Future
  8. Acknowledgements
  9. References
  10. Biographies

Biographies

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Brownian motion in a gas
  5. 3 Brownian motion in a liquid
  6. 4 Effects of detection noise on studying Brownian motion
  7. 5 Future
  8. Acknowledgements
  9. References
  10. Biographies
  • Image of creator

    Tongcang Li received his Ph.D. degree from the University of Texas at Austin in 2011. During his Ph.D. study under the supervision of Mark Raizen, he experimented on Bose-Einstein condensates and Brownian motion of particles suspended in liquids, air and vacuum. He is currently a postdoctoral fellow at the University of California, Berkeley. He recently proposed to create space-time crystals experimentally with trapped ions.

  • Image of creator

    Mark G. Raizen is a professor and holds the Sid W. Richardson Foundation Regents Chair of Physics at the University of Texas at Austin where he also received his Ph.D. degree in 1989. He recently developed general methods to control the motion of atoms and molecules, and studied Brownian motion with unprecedented precision.