Seminar I - 1.letnik, II. stopnja:

Brownian motion on short time scales

Author: Bostjan Jencic

Mentor: prof. dr. Rudolf Podgornik

Ljubljana, May 2013

Abstract

The seminar presents Brownian motion at short timescales, when the self-similarity ofrandom Brownian motion is expected to break down, and be replaced by so called ballisticmotion, where the particle’s momentum cannot be neglected. Short time Brownian motionin a gas is presented in theory and experimentally for a free particle and for opticallytrapped particle. The calculated mean square displacement (MSD) for a free particle iscompared to the well known MSD at long time scales. Theory of the Brownian motionin a liquid is then presented seperately, because the momentum of the surrounding liquidmust also be taken into account and therefore the results differ from the case in a gas.

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Contents

1 Introduction 2

2 Brownian motion at long and short time scales 3

2.1 Brownian motion in a gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Theory-free Brownian particle . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Theory of optically trapped Brownian particle in a gas . . . . . . . . . . 5

2.1.3 Experimental observation of Brownian particle in air . . . . . . . . . . . 5

2.2 Brownian motion in a liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Theory of a free particle in a liquid . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Theory of an optically trapped microsphere in a liquid . . . . . . . . . . 10

2.2.3 Experimental observation of the transition from ballistic to diffusiveBrownian motion in a liquid . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Conclusion 13

1 Introduction

Brownian motion is apparently random movement of particles suspended in a liquid or a gas,because of their interactions with other particles. It was first observed in 1827, when botanistRobert Brown found, that pollen grains (with dimensions of about 5µm ) moved throughthe water, although he was unable to determine the mechanisms that caused this motions.After almost a century, in 1905, Albert Einstein presented first fully developed theory aboutBrownian motion, that based on statistical mechanics. Einstein’s theory explained, that pollengrains were moved by water molecules with linear momentum, which, due to large frequencyof collisions, was enough to significantly change the course of pollen grains, although they weremuch bigger than molecules of water. Einstein’s theory was the first evidence that atoms andmolecules really exist, and was later experimentally proven by Jean Perrin in 1908, who wasawarded the Nobel Prize in Physics in 1926 [1].

Figure 1: Brownian movement reproduced from the book of Jean Perrin. Three tracings ofthe motion of colloidal particles (radius 0.53µm) are displyed as seen under the microscope.Successive positions every 30 seconds are joined by straight lines [1].

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2 Brownian motion at long and short time scales

Brownian motion was first explained by Einstein, who proposed, that apparently randommovement of observed particle is caused by thermal motion of surrounding fluid molecules.The long time scale based Einstein theory predicts, that the mean-square displacement (MSD)of a free Brownian particle is proportional to the product of diffusion constant D and elapsedtime t.

⟨(x(t)− x(0))2

⟩=⟨∆x2

⟩= 2Dt (1)

D =kBT

6πηR(2)

The diffusion constant is composed of thermal energy of a fluid (kBT ), and of the Stokesfriction coefficient (6πηR), where R is the radius of a free particle, and η is the viscositycoefficient of a fluid [2].

This result is correct only at long time scales, where statistical approach can be used andparticle momentum can be averaged to 0. Long time trajectories of Brownian particles areclassic examples of fractals, which means, that they are continuous everywhere, and notdifferentiable anywhere [3]. Because function x(t) is not differentiable in case of Brownianparticle, we cannot determine its velocity at any given time. According to Einstein’s theory,the mean velocity could be calculated from equation 1:

〈v〉 =

√〈∆x2〉∆t

=

√2D

t. (3)

We can see, that the mean velocity diverges as t approaches 0, and therefore cannot be usedas an appropriate description of movement of a Brownian particle at short time scales.

To define term ”short time”, we introduce characteristic time τp = M6πηR , where M is the mass

of the Brownian particle. For t τp, we can assume, that t is really short, and Brownianparticle is expected to be dominated by its inertia, so its trajectory cannot be self similar.This, so called ballistic Brownian motion, is much different than Einstein’s theory based”diffusive Brownian motion”.

2.1 Brownian motion in a gas

2.1.1 Theory-free Brownian particle

Since the lower values of viscosity and density of a gas (compared to a liquid), relaxation timeτp is much bigger, and therefore experimental observations are easier. Lower density alsoallows us to neglect the inertia effects of the gas and focus only on inertia of the Brownianparticle.

At ordinary conditions (pressure 100kPa, temperature 300K), mean free path of molecules inair is approximately 68 nm [4]. About 1µm large microsphere suspended in a gas, collides withsurrounding molecules with a frequency of about 1016Hz, if ordinary conditions are present.Due to much larger mass of the microsphere, only a large amount of collisions can make asignificant change of microsphere’s velocity. This is reflected by relatively large relaxationtime, τp = 6µs. For times much smaller than τp, but still much larger than time between twoconsecutive collisions, we can describe the dynamics of a Brownian particle with Langevinequation [5]:

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Md2x

dt2+ γ

dx

dt= Ftherm(t), (4)

where γ = 6πηR is the Stokes friction coefficient, and

Ftherm(t) =√

2kBTγς(t) (5)

is the Brownian (thermal) stochastic force. ς(t) is a normalized white noise process, withfollowing properties:

〈ς(t)〉 = 0, (6)

⟨ς(t)ς(t′)

⟩= δ(t− t′). (7)

The solution of such differential equation gives us the MSD for a Brownian particle [6]:

⟨∆x2

⟩=

2kBT

Mτ2p (t

τp− 1 + e−t/τp). (8)

A closer look at both limits shows us the difference between ballistic and diffusive Brownianmotion. For t τp, the result is equal to the one predicted by Einstein’s theory. But fort τp, we can see, that expression of MSD significantly differs from the previous one

⟨∆x2(t τp)

⟩=kBT

Mt2. (9)

We can also see, that velocity autocorrelation function is not 0 as in the diffusive case, but itis [6]:

〈v(t)v(0)〉 =kBT

Me−t/τp . (10)

Although these equations are originally derived for an ensemble of particles, the ergodictheorem dictates, that they are also valid for measurements of a single particle taken over along time.

The more precise calculation of relaxation time τp can be done with help of kinetic theory.Value of Γ0 = 1

τp, called damping coefficient, can be obtained by equation [7]:

Γ0 =6πηR

M

0.619

0.619 +Kn(1 + cK), (11)

where Kn = sR is Knudsen number. Here, s is the mean free path of molecules in a gas.

ck=0.31Kn

0.785+1.152Kn+Kn2 is a small positive function of Kn. We can see, that at high pressures,

where mean free path and consequencially Kn is small, Γ0 approaches the value of 6πηRM , as

previously obtained.

In case of ballistic Brownian motion, where velocity can be measured as v = ∆x∆t , the amplitude

of velocity is determined by the temperature of environment. From kinetic theory we know,that distribution of the velocity of a particle (molecule) in thermal equilibrium is

fv(vi) =

√M

2πkBTexp(− Mv2

i

2kBT) (12)

where i can represent any direction in carthesic x,y,z coordinate system.

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2.1.2 Theory of optically trapped Brownian particle in a gas

Due to experimental reasons, theory of optically trapped microsphere in a gas will also bepresented. Brownian particle can be trapped by using optical tweezers. For relatively smalldiscplacements, we can approximate the optical tweezers’ effect as that of a harmonic poten-tial. Thus, the obtained differential equation is:

d2x

dt2+ Γ0

dx

dt+ Ω2x = Λς(t). (13)

Here Ω =√

km represents the natural angular frequency of the trapped microsphere, when

no damping is present, and Λ =√

2kBTΓ0/M is the scale for white noise Brownian force.

The effective damped oscillator frequency is ω1 =√

Ω2 − Γ20/4. The harmonic system can

be called underdamped, when ω1 is real (Ω > Γ0/2), critically damped when ω1 = 0, andoverdamped when frequency ω1 is imaginary.

The MSD of optically trapped Brownian particle in a gas is [8]:

⟨∆x2

⟩=

2kBT

MΩ2[1− e−t/2τp(cos(ω1t) +

sin(ω1t)

2ω1τp)]. (14)

The normalized velocity autocorrelation function of such particle is [8]:

〈v(t)v(0)〉v2

= e−t/τp(cos(ω1t)−sin(ω1t)

2ω1t) (15)

We can see, that both MSC and velocity autocorrelation function oscilate for an underdampedsystem (real ω1).

2.1.3 Experimental observation of Brownian particle in air

Brownian particle in air is much easier to observe than in liquid, due to lower viscosity of air.The main problem of such experiment is, that observed microsphere falls under the influenceof gravity. Because of that, it is nearly impossible to deal with a free particle. This problemcan be solved by using optical tweezers to simultaneously trap and monitor microsphere inair and vacuum, allowing long time measurements of its motion [9].

One way to observe optically trapped microsphere is with a laser, which is used both fortrapping and detection. This dual usage is achieved by adding a polarizing beam splittercube to reflect one of the trapping beams for detection. A balanced detector is also neededalong with sharp edges of a mirror in order to split one beam in to two.

The trap is formed inside a vacuum chamber by two counter-propagating and orthogonalypolarized laser beams focused to the same point by two identical aspheric lenses. For max-imizing the trapping frequency, we keep the power of the first laser constant while we tunethe power of the second one. If observed microsphere deviates from the center of a trap, itdeflects the beams of both lasers. One deflection can be measured, and is good indication ofmicrosphere’s position. This beam is split by a mirror with a sharp edge.

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Figure 2: Simplified scheme of experiment. The s-polarized beam is reflected by a polarizingbream splitter cube after it passes through a trapped microsphere inside a vacuum chamber.Then it is split by a mirror with a sharp edge in order to detect. The p-polarized beam passesthrough the cube [10].

The experiment was done at two different pressures, 2.75kPa and 99.8kPa. The satisfyingrelaxation times for those two pressures are τp1 = 147µs and τp2 = 48µs. At T=300K, themean free path for those two pressures is approximately 68 nm and 2.47µm.

Results of the experiment fit perfectly with theory previously described. First figure showsus position and instantaneous velocity of microsphere in relation to time. We can see, thatat higher pressure, instantaneous velocity changes more rapidly due to lower relaxation time.

Figure 3: 1D trajectories of a 3µm diameter microsphere trapped in air at two differentpressures, 99.8kPa (A) and 2.75kPa (B). The corresponding instantaneous velocities of theBrownian particle are shown in C (99.8kPa) and D (2.75kPa) [10].

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Figure 4 shows us mean square displacement in relation to time. The measured also MSD’sfit excellently with theoretically predicted behaviour over three decades of time. It should bealso noted, that slopes differ from those, predicted by Einstein’s diffusive theory.

Figure 4: Measured MSD for microsphere for 99.8kPa (red square) and 2.75kPa(blue square).Solid lines represent theoretical predictions for MSD. Dashed lines are predictions of Einstein’stheory of free Brownian motion in the diffusive regime [10].

Figure 5 shows us the measured instantaneous velocity distribution. Measurement agrees verywell with predicted Maxwell-Boltzmann distribution. As expected, the velocity distributionis also independent of pressure.

Figure 5: The one dimensional distribution of the measured instantaneous velocities of mi-crosphere. For both pressures, 4 milion measurements were used in statistics. Black solid linerepresents theoretical Maxwell-Boltzmann distribution [10].

Velocity autocorrelation function was also measured for both pressures. This measurementcan be an accurate method to obtain τp and ω1 for a certain pressure. As shown in figure 6,

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velocites at short time intervals are very correlated but their correlation decays exponentiallywith time.

Figure 6: The normalized velocity autocorrelation functions of microsphere for 99.8kPa (redsquare) and 2.75kPa (black circle). Solid red and black lines represent theoretical predictionof VACF at those pressures [10].

2.2 Brownian motion in a liquid

There are some important differences between Brownian motion in liquid and in a gas. Themost important one is, that when dealing with liquid, we must take hydrodynamic effects ofthe liquid into account. Besides the inertia of the Brownian particle itself, the inertia of thesurrounding liquid is also important. Brownian particle’s motion will cause numerous long-lived vortices, which will affect the motion of particle itself. This, so called hydrodynamiceffect of a liquid, is actually the main contribution to dynamics of particle at short time scales.

2.2.1 Theory of a free particle in a liquid

Dynamics of a free Brownian particle in a liquid can also be studied with help of Langevinequation:

Mpx(t) = Fstokes(t) + Fthermal(t). (16)

Assuming that the velocity of the Brownian particle is always significantlly smaller than thespeed of sound in a liquid, the fluid motion can be described by the linearized incompressibletime-dependent Navier-Stokes equation.

Because of the hydrodynamic effects, Stokes friction force, derived from Navier-Stokes equa-tion, becomes more complicated[11]:

Fstokes(t) = −γx− 1

2Mf x− 6R2√πηρf

tˆ

−∞

(t− t′)−1/2x(t′)dt′. (17)

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Besides the regular term −γx, two new terms are added.

The first one, −12Mf x , shows us, that the friction force is also related to the acceleration of

the microsphere and Mf = 43πR

3ρf is the mass of displaced liquid (ρf is the density of theliquid). Due to this acceleration term, the new effective mass of the microsphere, is equal tothe sum of the real mass and half of the mass of displaced liquid[11]:

M∗ = Mp +1

2Mf . (18)

According to new effective mass, we must also modify the energy equipartiotion theorem:

1

2M∗

⟨v2⟩

=1

2kBT. (19)

The second new term, −6R2√πηρf´ t−∞(t − t′)−1/2x(t′)dt′, represents the hydrodynamic re-

tardation effects of the liquid. As can be seen from the integral, the acceleration of microspherefrom all past times contributes to the friction force at present. However, the contribution oftimes t′ close to −∞ is significantly smaller than the contribution of times close to t, becauseof term (t− t′)−1/2.

The Brownian force is also different than in a gas, and it is not a white noise anymore. Thehydrodynamic memory of the liquid will affect both the termal and the frictional force, sothose two forces will become directly related. The new correlation of the thermal force is nota delta function anymore, but it is proportional to t−3/2 [13]:

〈Ftherm(t)Ftherm(0)〉 = −γkBT√τf4πt−3/2 (20)

The whole Langevin equation is therefore:

M∗x(t) = −6πηRx(t)− 6R2√πρfηtˆ

−∞

x(t′)dt′√t− t′

+ Ftherm(t). (21)

The normalized VACF, obtained from Langevin equation for a free particle in a liquid is [12]:

v(t)v(0)

kBT/M∗=α+e

α2+terfc(α+

√t)− α−eα

2−terfc(α−

√t)

α+ − α−, (22)

where

α± =3

2

3±√

5− 36τp/τf√τf (1 + 9τp/τf )

. (23)

τp = Mp/(6πηR) and τf = R2ρf/η are momentum relaxation times of the particle and of aliquid.

As t approaches 0, VACF behaves as exp(−b√t/τf ) , where

b =18√

π(1 + 2ρp/ρf ), (24)

and for long times (t→∞), VACF is proportional to 1/t3/2.

As can be seen, the VACF approaches 1 at short time scales as exp(−b√t/τf ) , rather than

exp(−t/τp) (as in a gas), so the dynamics of the particle is dominated by the hydrodinamiceffects of the liquid, which is very different from the case in the air.

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2.2.2 Theory of an optically trapped microsphere in a liquid

A harmonic force provided by the optical trap is Ftrap = −kx, where x is a small microsphere’sdisplacement. The natural angular frequency of the trap is Ω =

√k/Mp.

Langevin equation of a trapped microsphere in an incompressible liquid is therefore [13]:

M∗x(t) = −kx(t)− 6πηRx(t)− 6R2√πρfηtˆ

−∞

x(t′)dt′√t− t′

+ Ftherm(t). (25)

As previously, first term on the right side represents the harmonic force, the second one isassociated with regular Stokes friction, the third one represents the hydrodynamic retardationeffects of the liquid and the fourth is the Brownian force.

Table 1: Characteristic time scales of an optically trapped silica microsphere in water at 293K[9].

In case of optically trapped microsphere in liquid, it’s MSD is [13]:

⟨∆x2

⟩=

2kBT

k+

2kBT

M∗[

ez21terfc(z1

√t)

z1(z1 − z2)(z1 − z3)(z1 − z4)+

ez22terfc(z2

√t)

z2(z2 − z1)(z2 − z3)(z2 − z4)+

+ez

23terfc(z3

√t)

z3(z3 − z1)(z3 − z2)(z3 − z4)+

ez24terfc(z4

√t)

z4(z4 − z1)(z4 − z2)(z4 − z3)] (26)

The coefficients z1, z2, z3 and z4 are obtained from the equation

(τp +1

9τf )z4 −√τfz3 + z2 +

1

τk= 0 (27)

where τk = 6πηR/k is the relaxation time related to the optical trap. When t goes to infinity,the MSD approaches to 2kBT/k.

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Figure 7: Calculated MSD for optically trapped microsphere in water with four differentdiameters. Water temperature is 293K. All MSD are limited by value of 2kBT/k. Slope 2shows the expected behaviour od the ballistic Brownian motion of a free particle [9].

We can also derive the normalized velocity autocorrelation function of a trapped microspherein a liquid [13]:

〈v(t)v(0)〉kBT/M∗

=z3

1ez21terfc(z1

√t)

(z1 − z2)(z1 − z3)(z1 − z4)+

z32ez22terfc(z2

√t)

(z2 − z1)(z2 − z3)(z2 − z4)+ .

+z3

3ez23terfc(z3

√t)

(z3 − z1)(z3 − z2)(z3 − z4)+

z34ez24terfc(z4

√t)

(z4 − z1)(z4 − z2)(z4 − z3). (28)

The limit of t → 0 shows us, that the VACF approaches the value kBT/M∗, instead of the

result predicted by the energy equipartition theorem kBT/Mp (where Mp is the mass of amicrosphere). This significant difference is a consequence of an assumption, that liquid isincompressible. For t ∼ tc, where tc = R/c is the time required for a sound wave to travela sphere radius (c is the speed of sound in the liquid), we need to consider the liquid ascompressible, and the expected behaviour of the Brownian motion is different.

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Figure 8: Calculated normalized VACF of an optcally trapped microsphere in water at 293Kfor four different microsphere diameters. For t ∼ tc(about 1ns in upper case), fluid is treatedas compressible (solid thin lines), and for t tc it is considered incompressible (solid thicklines). The intermediate regime is poorly understood. The dashed lines represent exponentialdecays for 3µm microsphere (τp = 1.0µs) [9].

VACF at t=0 does not approach 1, but 1 + Mf/2Mp because the normalization factor iskBT/M

∗ instead of kBT/Mp.

2.2.3 Experimental observation of the transition from ballistic to diffusive Brownianmotion in a liquid

Recently, the Brownian motion of a single particle in an optical trap in water was studied[14]. Both, MSD and VACF were measured. Figure 9 shows results of MSD’s for two silicamicrospheres with diameters of 1 and 2.5µm. It can be clearly seen, that at short time scales,microspheres move as free particles, and their slopes are close to the slope predicted by ballisticBrownian motion. After a long time, the MSD’s are limited because of the confinement ofthe optical trap. Figure 10 shows experimental results of the VACF for a 2µm particle and atheoretical fit with the equation 25. Curves obtained from the Langevin equation neglectinghydrodynamic effects of water are also displayed for comparison. For times shorter thanτf = 1µs, the VACF is smaller than the Langevin equations predict, because of the neglectedinertia of the fluid. For times longer than τf , the correlations are stronger due to the vortexesdeveloped in the fluid.

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Figure 9: Measured MSD for two different sized silica spheres. The red lines show the expectedMSD for ballistic Brownian motion [14]

Figure 10: Experimentally obtained normalized VACF for microsphere in an optical trap.The blue line shows the fitting with eq. 25, and the grey dashed and green lines show theVACF of free and optically trapped microsphere when inertia of the liquid is neglected [14].

3 Conclusion

Brownian motion is best described by numerous statistical properties, two of them beingmean square displacement (MSD) and velocity autocorrelation function (VACF). MSD andVACF as functions of time are very different for short and long time scales for all cases of

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Brownian motion either in a gas or in a liquid. Current experimental techniques allready allowscientists to observe Brownian motion at much shorter time scales than relaxation times τp,even in liquid. But instantaneous velocity of the Brownian particle was still not measuredin a liquid, only in a gas. The velocity measurement could be used to test the modifiedequipartition theorem for Brownian motion in a liquid (eq. 19) and the Maxwell-Boltzmannvelocity distribution. A measurement of the VACF>0.35 (even higher measuring frequenciesas to date) will deepen understanding of the hydrodynamic effects and compressibillity effectsof a liquid on Brownian motion.

References

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