Dynamic Light Scattering Experiment DLS University of Florida | Department of Physics PHY4803L | Advanced Physics Laboratory Objective Introduction A HeNe laser beam incident on a sample of Scattering experiments can provide a wealth micron-sized spheres suspended in water scat- of detailed information about the structural ters into a random pattern of spots of varying and dynamical properties of matter. Dynamic size, shape, and intensity. The pattern results light scattering, in particular, has important from the coherent superposition of the outgo- applications in particle and macromolecule ing waves scattered from the spheres. Because sizing and brings in another important topic in the spheres are in constant Brownian motion, physics: Brownian motion. We begin with a the pattern randomly changes in time. A pho- brief discussion of Brownian motion and dy- todetector is placed in the pattern and the namic light scattering before describing the random fluctuations in the light intensity are apparatus and measurements. measured and analyzed. The signal's autocor- relation function and power spectrum are com- puted and used to verify the Brownian motion Brownian motion and determine the spheres' diameter. Brownian motion refers to the random diffu- sive motion of microscopic particles suspended in a liquid or a gas. This motion was first References studied in detail by Robert Brown in 1827 when he observed the motion of pollen grains Clark, Lunacek, and Benedek A Study of in water through his microscope. More sys- Brownian Motion Using Light Scattering, tematic studies found the motion depends on Amer. J. of Phys. 38, 575 (1970). particle size, liquid viscosity and tempera- ture and around 1905 Albert Einstein and M. Daniel T. Gillespie The mathematics of Smoluchowski independently connected Brow- Brownian motion and Johnson noise, nian motion to the kinetic theory. Amer. J. of Phys. 64 225 (1996). Before considering Brownian motion, let's first recall certain aspects of the kinetic the- ory for the molecules of the suspension liq- Daniel T. Gillespie Fluctuation and dissipa- uid. These molecules are in constant ther- tion in Brownian motion, Amer. J. of mal motion having a Maxwell-Boltzmann ve- Phys. 61 1077 (1993). locity distribution. This distribution gives the DLS 1 DLS 2 Advanced Physics Laboratory probability for the molecule to have a velocity Numerical solutions to the motion of a par- component in the x-direction between vx and ticle typically begin with Newton's second law vx + dvx as cast in the form: s m − 2 mvx=2kB T dr(t) = v(t)dt (6) dP (vx) = e dvx (1) 2πkBT 1 dv(t) = F(t)dt (7) M where m is the molecular mass, kB is Boltz- mann's constant, and T is the temperature. where M is the particle mass, r(t) is its po- Of course, analogous expressions apply to the sition, v(t) is its velocity, and F(t) is the net y- and z-components of velocity. Equation 1 force on the particle. One chooses some small is a Gaussian probability distribution with a but finite dt over which r(t) and v(t) can be mean of zero and a variance of kBT=m. assumed constant. F(t), which may depend We will use a shorthand notation on r and v, is evaluated and the right sides of Eqs. 6 and 7 are calculated. With the left 2 N(µ, σ ) (2) sides defined by to express a Gaussian distribution of mean µ dr(t) = r(t + dt) − r(t) (8) and variance σ2 and, for example, the equation dv(t) = v(t + dt) − v(t) (9) ! kBT vx = N 0; (3) the right-side values are then added to the m values r(t) and v(t) to obtain updated values will be a shorthand notation expressing that r(t+dt) and v(t+dt) at a time dt later. Start- the x-component of velocity for a suspension ing from given initial conditions for r(0) = r0 molecule is a sample from the probability dis- and v(0) = v0 at t = 0, the process is repeated tribution of Eq. 1. to obtain future values for r(t) and v(t) at dis- The equipartition theorem states that each crete intervals. As we will see, this modeling degree of freedom must have an average en- of the equations of motion is particularly ap- ergy of kBT=2. For the translational degree of propriate for Brownian motion. freedom in the x-direction this implies: The motion is said to be deterministic when D E F(t) can be precisely determined from the val- 1 2 1 ues of r(t), v(t), and t. For example, in a colli- m vx = kBT (4) 2 2 sion between two particles with a known inter- where the angle brackets hi indicate tak- action (such as the Coulomb or gravitational ing an average over the appropriate proba- force) F(t) is deterministic and the motion is 1 bility distribution. For example, with the quite predictable. Maxwell-Boltzmann probability distribution For Brownian motion, F(t) arises from the continual collisions of suspension molecules for vx (Eq. 1), s against the particle. Each interaction with a D E Z 1 m − 2 suspension molecule during a collision delivers 2 2 mvx=2kB T vx = vxe dvx (5) −∞ 2πkBT 1Deterministic does not always mean predictable. Some perfectly precise forms of F(t) lead to chaotic h 2i which gives vx = kBT=m, and is clearly con- solutions that cannot be predicted far into the future sistent with Eq. 4. at all. June 14, 2012 Dynamic Light Scattering DLS 3 an impulse to the particle from a Gaussian distribution of mean µ = Nµi 2 2 Z and variance σ = Nσi . Ji = Fi(t)dt (10) Each cartesian component of Ji can be as- sumed to be a random number from some (un- where Fi(t) is the force on the particle and the known) distribution and thus the central limit integral extends over the duration of the col- theorem applies to each component of Eq. 11. lision. The individual impulses Ji vary in size Remember, v(t) and r(t) do not change sig- and direction depending on the speeds and an- nificantly over the interval dt; the probability gles involved in the collision. Velocities vary distributions for the components of Ji arise according to the Maxwell-Boltzmann distribu- from the distribution of velocities for the sus- tion and average around 600 m/s for room pension molecules and from the distribution of temperature water. Collisions are short and collision angles. Moreover, because the num- frequent, occurring around 1019 times per sec- ber of collisions N over a time interval dt will ond for a 1 µ particle in water. The random- be proportional to dt, the central limit theo- ness of the individual collisions leads to a net rem implies that each component of F(t)dt will force that includes random components and be a random number from a Gaussian distribu- the force and motion are said to be stochas- tion having a mean and variance proportional tic. The motion of a single particle is unpre- to dt. dictable and only probabilities or average be- Shortly after Einstein's work on the subject, havior can be determined. Paul Langevin hypothesized that F(t)dt can Because of the high collision frequency, we be expressed can choose a time interval dt short enough that F(t)dt = −αv(t) dt + F(r)(t) dt (12) r(t) and v(t) do not change significantly, yet long enough to include thousands of collisions. The viscous drag force −αv, opposite in di- Over such an interval, the value of F(t)dt in rection and proportional to the velocity, had Eq. 7 would properly be the sum of all im- already been investigated by Stokes, who pulses delivered during the interval dt showed that the drag coefficient for a sphere X of diameter d in a suspension of viscosity η is F(t)dt = Ji (11) given by i α = 3πηd (13) With enough collisions, the central limit the- F(r)(t) is the random part of the collisional orem can be used to draw important conclu- force, which Langevin successfully character- sions about the form of F(t)dt even though ized and showed how it was responsible for detailed knowledge of individual impulses is Brownian motion. lacking. Keeping in mind that any random number The central limit theorem states that the from a distribution with a mean µ and vari- sum of many random numbers will always be ance σ2 can be considered as the sum of the a Gaussian-distributed random number. More mean and a zero-mean random number having specifically, it states that if each of the individ- a variance σ2 ual random numbers are from a distribution N(µ, σ2) = µ + N(0; σ2) (14) (which need not be Gaussian) having a mean 2 µi and variance σi , then the sum of N such allows one to see how Eq. 12 is related to random numbers will be a random number Eq. 11 and the central limit theorem. Each June 14, 2012 DLS 4 Advanced Physics Laboratory cartesian component of the −αv dt term in chosen small enough that r(t) and v(t) make Eq. 12 is the mean of the sum in the cen- only small changes during the interval. How- tral limit theorem applied to that component ever, dt must not be made too small because of Eq. 11. With the means accounted for roundoff and other numerical errors occur with by the −αv dt term, each component of the each step.