Brownian Motion Experiment BM University of Florida | Department of Physics PHY4803L | Advanced Physics Laboratory By Robert DeSerio & Stephen Hagen temperature dependence of the Brownian mo- tion of polystyrene microspheres, Am J. Phys. 3 Objective 75 111-115 (2007). P. Nakroshis, M. Amoroso, J. Legere, C. A microscope is used to observe the motion Smith, Measuring Boltzmanns constant using of micron-sized spheres suspended in water. video microscopy of Brownian motion, Am. J. A digital video camera captures the motion Phys. 71 568-573 (2003).4 which is then analyzed with a particle track- P. Nelson, Biological Physics: Energy, In- ing program to determine the path of indi- formation, Life, W. H. Freeman (2003). vidual spheres. The sphere's random Brow- P. Pearle et al., What Brown saw and you nian motion is analyzed with a spreadsheet can too, Am. J. Phys. 78 1278-1289 (2010).5 to verify various theoretical predictions. The D.E. Smith, T.T. Perkins, S. Chu, Dy- dependence of the particle's displacements on namical scaling of DNA diffusion coefficients, time as well as various physical parameters Macromolecules 29 1372-1373 (1996).6 such as the temperature, the suspension liq- uid's viscosity and the sphere diameter is also explored. Diffusion of dye-labeled DNA Brownian motion molecules is also studied. Brownian motion refers to the continuous, random motion of microscopic particles that References are suspended in a fluid. Robert Brown in 1827 described such motion in micron-sized M.A. Catipovic, P.M. Tyler, J.G. Trapani, and particles that were released from the pollen Ashley R. Carter, Improving the quantifica- grains of Clarkia pulchella, a flower that had tion of Brownian motion, Am. J. Phys. 81 been discovered by Lewis and Clark a few 1 485-491 (2013). years earlier. Although the biological origin of Daniel T. Gillespie, The mathematics of the particles at first suggested that this motion Brownian motion and Johnson noise, Am. J. had something to do with life, Brown quickly 2 Phys. 64 225 (1996). realized that all sorts of inorganic particles ex- D. Jia, J. Hamilton, L.M. Zaman, and A. Goonewardene, The time, size, viscosity, and 3http://dx.doi.org/10.1119/1.2386163 4http://dx.doi.org/10.1119/1.1542619 1http://dx.doi.org/10.1119/1.4803529 5http://dx.doi.org/10.1119/1.3475685 2http://dx.doi.org/10.1119/1.18210 6http://pubs.acs.org/doi/abs/10.1021/ma951455p BM 1 BM 2 Advanced Physics Laboratory hibited the same motion. Granules of carbon Kinetic theory soot or ancient stone, when suspended in wa- ter, executed ceaseless motion with complex, Before considering Brownian motion, we apparently random trajectories. Although the should first recall certain aspects of the kinetic rate of motion depends on the size of the par- theory for the molecules of the fluid. These ticles and the temperature and viscosity of the molecules are in constant interaction with all fluid, the phenomenon is universal to particles other molecules, which together form a heat in a fluid. bath at temperature T . The equipartition the- orem of statistical physics requires that the kinetic energy in each spatial component of the molecular velocity has an ensemble aver- age value of kBT=2. For the x-direction this Brown himself did not understand the phys- implies: ical cause, but Einstein recognized that Brow- 1 1 m v2 = k T (1) nian motion could be completely explained in 2 x 2 B terms of the atomic nature of matter and the Here m is the molecular mass and T is the kinetic nature of heat: the fluid is composed temperature. The angle brackets hi indicate of molecules that are in continuous motion at an ensemble average, i.e. the average over a any finite temperature. A larger particle in the large population of molecules. The ensemble fluid is subject to frequent collisions that de- average may be calculated if the probability liver numerous small, random impulses, caus- distribution for v is known. ing the larger particle to drift gradually but x What is the probability distribution for the irregularly through the fluid. In one of his fa- velocity components? Each of the three spa- mous 1905 papers, Einstein showed that the tial components of the molecular velocity is rate of Brownian motion is directly related distributed according to a Boltzmann distri- to the microscopic Boltzmann constant k , B bution in the kinetic energy mv2=2 associated which sets the scale for the kinetic energy with that velocity component. This leads to (∼ k T ) carried by a water molecule at tem- B the so-called Maxwell Boltzmann distribution perature T . k is related to the macroscopic B for the probability that the molecule will have gas constant R = NAkBT by a factor of Avo- a velocity between vx and vx + dvx gadro's number NA. As R is easily found in a benchtop experiment, the measurement of r m 2 k reveals N . Brownian motion is a direct dP (v ) = e−mvx=2kB T dv (2) B A x 2πk T x link between two very different size scales in B physics: it originates in the microscopic mo- The square root prefactor is required for nor- tion of atoms, molecules and the tiny scale malization of their thermal energy, and it is observable as macroscopic motion that can be measured Z 1 dP (vx) = 1 (3) at the bench of any reasonably well equipped −∞ laboratory. Einstein's work paved the way for Perrin's measurements of Brownian mo- Equation2 is a Gaussian probability distribu- tion, which provided compelling support for tion with a mean of zero and a variance of the atomic theory and earned the 1926 physics kBT=m. Of course, analogous expressions ap- Nobel prize. ply to the y- and z-components of velocity. December 28, 2015 Brownian Motion BM 3 The Maxwell-Boltzmann probability distri- bution for vx (Eq.2) obeys the equipartition theorem because the average is r 1 m Z 2 2 2 −mvx=2kB T vx = vxe dvx (4) 2πkBT −∞ 2 which leads to hvxi = kBT=m, consistent with Eq.1. For water molecules at room tempera- ture the average molecular speed is roughly 600 m/s. Consequently there is plenty of mo- mentum in the fluid molecules that collide with a small particle that is suspended in the fluid. Consider a particle with mass M, suspended in the fluid at r(t) and moving with velocity Figure 1: The molecules of the surrounding fluid v(t). The particle is subject to a net force F(t) undergo frequent collisions with a Brownian par- from the fluid. We can analyze the motion ticle (shown as large circle), delivering small ran- over a time interval dt that is sufficiently short dom impulses Ji. There are many such impulses in even a short interval dt. that r(t) and v(t) can both be considered very nearly constant. For a Brownian particle, it is convenient to analyze the motion by casting ues of r(t), v(t), and t. For example, in a colli- Newton's second law in the form: sion between two particles with a known inter- action (such as the Coulomb or gravitational dr(t) = v(t)dt (5) force) F(t) is deterministic and the motion is 1 quite predictable.7 dv(t) = F(t)dt (6) M In Brownian motion, F(t) is created by the very frequent collisions between the suspended F(t), which may depend on r and v, is eval- particle and the molecules of the surrounding uated and the right sides of Eqs.5 and6 are medium, about 1019 collisions per second for calculated. With the left sides defined by a 1 µm particle in water. Consider the force dr(t) = r(t + dt) − r(t) (7) Fi(t) that acts on the Brownian particle dur- ing its collision with one individual molecule dv(t) = v(t + dt) − v(t) (8) from the medium. That one collision delivers the right-side values are then added to the an impulse to the Brownian particle values r(t) and v(t) to obtain updated values Z r(t+dt) and v(t+dt) at a time dt later. Start- Ji = Fi(t)dt (9) ing from given initial conditions for r(0) = r0 and v(0) = v0 at t = 0, the process is repeated where the integral extends over the dura- to obtain future values for r(t) and v(t) at dis- 7Deterministic does not always mean predictable. crete intervals. Some perfectly precise forms of F(t) lead to chaotic The motion is said to be deterministic when solutions that cannot be predicted far into the future F(t) can be precisely determined from the val- at all. December 28, 2015 BM 4 Advanced Physics Laboratory tion of the collision (Figure1). The indi- a Gaussian-distributed random number with 2 2 vidual impulses Ji vary in magnitude and di- mean µ = Nµi and variance σ = Nσi . rection because the velocities of molecules in Each cartesian component of Ji is a random the medium vary according to the Maxwell- number from an (unknown) distribution and Boltzmann distribution. Consequently the di- thus the central limit theorem applies to each rection and magnitude of Fi(t) are highly vari- component of Eq. 10. Remember, v(t) and able. This variable impulse adds a random r(t) do not change significantly over the in- component to the net force on the particle. terval dt; the probability distributions for the Both the force and the particle's motion are components of Ji arise from the distribution of said to be stochastic, and the motion of a velocities for the colliding molecules and from single particle is unpredictable.