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Physics

A STUDY OF DIFFUSION BY LIGHT SCATTERING

I. INTRODUCTION .....................................................................................................................................................

II. THEORETICAL CONSIDERATIONS ................................................................................................................

III. MEASUREMENTS................................................................................................................................................

IV. DATA ANALYSIS................................................................................................................................................

I. Introduction

In 1827 an English botanist, Robert Brown, noticed that pollen grains in water perform peculiarly erratic movements. Some simple experiments convinced him that the phenomenon, now called Brownian motion, was physical, not biological, and could be observed for any small particle freely suspended in a fluid (gas or liquid). With the development of kinetic theory in the late nineteenth century it became evident that Brownian motion was due to the continual buffeting of the suspended particle by the atoms of the surrounding fluid. The first reasonably satisfactory theory was advanced by A. Einstein in 1905 and subsequently refined by many authors. It would obviously be very tedious and difficult to follow the paths of a large number of individual particles to study Brownian motion. Instead, one can use the time-varying intensity of light scattered from the suspension to gain some information about the average motions of the particles. The procedure is to create a mathematical model of the particle motion, calculate the light scattering that would result, and compare the calculations with experiment. The basic thermally-induced motions are now well enough understood that light scattering measurements are used to determine many properties of particles or large molecules in suspensions, including size and shape distributions. Interactions between particles will also modify the individual motions in characteristic ways, allowing one to determine some features of the inter-particle forces. It has even been possible to determine the number of living bacteria in a culture medium by detecting how many are swimming actively. The present experiment utilizes light scattering to examine the Brownian motion in a particularly simple system, a dilute suspension of identical polystyrene spheres. In the next section we present a simple model of our system and calculate the scattered intensity I ( t ). The intensity is expected to vary randomly with time, so we also develop a probabilistic description of such fluctuating signals. Succeeding sections describe the hardware and software needed for the experiment.

II. Theoretical Considerations

A. Model of the System We start by calculating the intensity of light scattered by a collection of N identical spheres of radius! a , illuminated by a monochromatic plane wave. The spheres are located at coordinates r (^) j ( t ) relative to a fixed origin in the scattering volume, as shown in Fig. 1. The total electric

field at a distant detector is the sum of the field radiated by each sphere

! E ( t ) =

!

E (^) j j = 1

N ! ( t )^ =^

!

E 0 '

j = 1

N ! exp^ i q! •! r [ (^) j "^ i^ # t ] (1)

In the simplest approximation, valid when ( !/ a ) 2 » 1,

!

E 0 '^ depends only on the wavelength, particle size and other constants. The quantity q! •! r (^) j is the phase shift of the scattered wave

relative to a wave scattered from a particle at the origin. The scattering vector q! is defined as the difference of the wave vectors of the incident and scattered light:

q^! =! k 0!^

!

k (^) s (2)

with magnitude

q = 4! n " sin^ # 2

(3)

where n is the index of refraction of the medium,! is the incident wavelength and " is the scattering angle. (It is not coincidental that Eq. 3 resembles the Bragg condition for diffraction

from gratings or crystals.) The total scattered intensity is simply

!

E( t ) 2 or

k (^) o

!

k (^) s

!

! r (^) j

"

Fig. 1 Light of wave vector

!

k0 scattering at angle " from a single particle located at! r (^) j.

I ( t ) = E 0 '^

2 exp (^) [ i q! • (! r (^) j!! r (^) k )] j , k

" (4)

where the j,k sum runs over the N particles. Evidently I ( t ) will vary as the particles move, changing! r ( t ) randomly. In order to describe the motion, we define a probability density P (! r' , t ' ;! r, t ). This is the probability of finding a given

particle in a small volume at! r' at time t ', given that it was at! r at time t. Since we assume that the particles do not interact, P (! r' , t ' ;! r, t ) must depend only on the coordinates of the chosen

particle, and must be the same function for all the particles. Further, there is nothing about the system which picks out a special origin of coordinates or time, so the function must depend only on spatial and time differences

!

R =! r' !! r and! = t' - t. A non-rigorous derivation of P

!

( R ,^ !;0, 0) proceeds as follows. Consider a region of fluid in

which the number of particles per unit volume n ( ! r ) varies with! r , as suggested by Fig. 2. The motions of the particles are random, so the number of particles crossing any plane in the fluid is proportional to the number density at that point. In Fig. 2, then, the number crossing in the + x direction is proportional to n ( x ) while the number crossing in the - x direction is proportional to n ( x + dx ). Defining

!

J n as the net number of particles crossing the plane per unit area per unit time,

one finds that

! J n =! D

!

" n (5)

The parameter D which makes this an equality is called the diffusion coefficient. It must also be true that the number of particles in a fixed volume changes only because particles enter or leave the volume at the boundaries (conservation of particles) so

n(x)

n(x + dx)

x^ Fig. 2 Net particle flux across a plane due to a concentration gradient.

! n ! t

+

!

" •

!

J n = 0 (6)

Putting these together leads to the basic equation for the variation in number density

! n ! t " D #^2 n = 0 (7)

Since the probability of finding a particle in a small volume is directly proportional to the number density, this leads to the equation for P

!

( R ,^ !;0, 0) :

! P

! t

" D #^2 P = 0 (8)

We want a solution to this equation with the initial condition that the particle is at! r = 0 at t = 0,

or P (0, 0;0, 0 ) !"(! r ). One can show by direct differentiation that

P

!

( R ,^ !;0, 0) =^ (^4 " Dt )#3/ 2^ exp^ [ #^ R^2 / 4 Dt ] (9)

is a solution of Eq. 8 and fits the required initial condition. The diffusion coefficient D can be related to other properties of the system by invoking the Einstein relation

D = k (^) BT μ (10)

The quantity μ is the mobility, the ratio of drift velocity to driving force when the particle is pushed through the fluid by an external force. (Note that this equation relates a thermal equilibrium quantity, D , to the non-equilibrium response to an external force. This sort of relationship is a fundamental property of many-particle systems.) The mobility can be calculated from classical hydrodynamics for spheres of radius a which are large compared to the surrounding molecules, with the result

vdrift = Fext / 6 !" a (11)

and therefore

D = k (^) BT / 6 !" a (12)

If the viscosity! of the solvent is known this will let us compare the measured and expected diffusion constants.

B. Analysis of the intensity fluctuations Each time we measure I ( t ) over a time interval 0! T we expect to get a different result because the particles will occupy different positions. There is, however, some statistical regularity in I ( t ) which is determined by P

!

( R ,^ !;0, 0). We will exploit this regularity by Fourier

analyzing I ( t ) and averaging the results over many trials to determine the frequency components present. A comparison of the measured and calculated frequency spectra will provide the desired test of our system model. To avoid convergence problems and to emulate the experiment we imagine sampling I ( t ) for a finite time T. The Fourier transform is then given by

F( I ) = I ( t ) e iwt !"

+"

dt^ =^ I ( t^ ) e^

i $ t 0

T

dt^ (13)

We will be interested in the "power spectrum", which is proportional to the magnitude-squared of the Fourier transform:

S (! ) = lim T "#

T

F( I ) 2 (14)

or, explicitly,

S ( !) = lim T "#

T

$$ I *( t ' ) I ( t^ ) e % i^!^ ( t '^ % t ) dt '^ dt^ (15)

The integral is simplified by letting t ' = t + " so that the integrations run over ", t. As noted above, averages must be independent of the origin of time, so we choose t = 0, making the integrand independent of t. Recalling that I ( t ) is sampled for a finite interval T , the effective range of the t -integral is 0! T and the limit is trivial, leaving

S ( !) = I *

• "

+"

(^ $) I (0) e

% i !$ (^) d $ (16)

This is the expression that must be averaged over particle positions. The formal average of S ( !) is written

S ( !) = P (

!

R ,^ ";0, 0)^ S (^!^ ) d^3

!

R (17)

where the integration is over the illuminated volume of the sample. It is advantageous to interchange the order of integration on d " and d^3

!

R to get

S ( !) = I *( ") I (0)

#$

+$ % e

i !" (^) d " (18)

Writing out the average I *^ (! ) I (0) gives

I *^ (! ) I (0) = E 0 '^

4 exp[" i q! • (! r l ( !) "! r m (! ))] exp[ i! q • (! r j (0) "! r k (0))] j , k

l , m

(19)

Using our assumption that the motions of different particles are independent allows us to factor most of the terms into a product of one or more single-particle averages exp(! i q! •! r ). Since

there is no preferred position, these single-particle averages are zero, eliminating most of the N^4 terms in Eq. 19. The only terms which survive are those containing two pairs of equal indices, which are:

l = m j = k N! N = N^2 terms l = k m = j m " l N! ( N -1) # N^2 terms (20) l = j m = k m " j N! ( N -1) # N^2 terms

The first set of terms is just <1! 1> = 1. The second group can be rewritten as

exp[! i q! •

!

R ( " )] exp[! 2 i q! •! r l (0)] 2 = 0 (21)

since

!

R (! ) is independent of! r. The third group is non-zero:

exp[! i! q • (! r l ( " )!! r l (0))] exp[ i! q • (! r m ( ")!! r m (0))] = exp[! i! q •

!

R ( " )]

2 (22)

The averaging integral can be done analytically, yielding

exp[! i! q •

!

R ( ")]

2 = exp(! 2 Dq^2 " ) (23)

Combining the terms we at last get

I *^ (! ) I (0) = N^2 E 0 '^

4

[^1 +^ exp("^2 Dq^2 !)] (24)

which can be substituted into Eq. 18 to give

S ( !) = N^2 E 0 '^

4 2 "#(! ) + 4 Dq

2 (2 Dq^2 ) 2 +! 2

$

%

&

'

(

)

(25)

This indicates that the power spectrum of I ( t ) consists of a delta function at! = 0, representing the average intensity, and a broad peak with a frequency width proportional to 2 Dq^2. There are several points of comparison between the measured and calculated power spectra. First, one can verify that < S ( !)> has the predicted form by fitting the expected function to the data and checking the "^2. The width, found in the fitting procedure, should vary as sin^2 ( #/2) as the scattering angle # is varied. Finally, the extracted value of D should be approximately equal to that estimated from Eq. 12. If one or more of these tests fail, it would tell us that the model must include other factors, such as particle interactions or multiple-scattering of the light. A numerical estimate of the expected width is useful at this point. The viscosity of water at 20oC is about 0.01 poise, which gives aD! 2.2 " 10 -13^ sec -1^. For laser light with $ = 632nm, an index of refraction n = 1.33 and # = 90o we find q! 1.8 " 105 cm-1^. A typical particle radius would be a = 0.1μm so 2 Dq^2 /2 %! 230 Hz (The 2 % is included because the apparatus is calibrated in Hz, not angular frequency.) Accordingly, we must plan to measure fluctuation rates up to several hundred hertz in order to adequately capture the input signal.

III. Measurements

The apparatus, shown schematically in Fig. 3, comprises a laser source, sample and holder, photo multiplier detector system and computer interface. Together, this equipment allows us to convert the intensity of light scattered from the sample at a specified angle into a form suitable for computer analysis.

A. General Arrangements The helium-neon laser produces about 2 mW output power at a wavelength of 632.8 nm. (Note that this is sufficient power to cause eye damage in the event of prolonged direct exposure. DO NOT LOOK INTO THE BEAM.) For maximum scattering efficiency the beam polarization must be normal to the (horizontal) scattering plane. Check that the "Caution" label is vertically above the output port, since this orients the internal mirrors of the laser for vertical polarization. A small lens is used to concentrate the beam at the center of the sample. The scattered light is collected by another lens mounted on an arm whose angular position can be set as desired. A photo multiplier tube is located behind an aperture placed one focal length beyond the collecting lens. The effect of this arrangement is to accept only light scattered within a narrow range of angles. An aperture placed near the sample tube minimizes the light from areas outside the sample volume. A polarizer and optical bandpass filter immediately in front of the detector ensure that only vertically polarized light at the laser wavelength can reach the tube. Together, these components also reduce the possibility that external light sources can damage the photo multiplier, but it is probably best to keep the HV off when the main room lights are on. Use the small work lights when you need to make adjustments with HV on. Samples are contained in cylindrical glass vials which fit into the supplied holder. The holder should be aligned so that the input beam follows a diameter of the cylinder. The samples consist of uniform polystyrene spheres dispersed in water and in methyl alcohol. (The samples actually

Laser

HV

sample

PMT I !V

I DC

Lowpass

"

Fig. 3 Schematic diagram of apparatus

contain a small amount of detergent, 0.1 wt% sodium dodecyl sulfate, and fungicide, 0.1 wt% sodium azide) Each sample is marked with the nominal sphere diameter and type of solvent. You will need to know the actual viscosity of the solvent at the measuring temperature, so note the room temperature when you take the data. Some pertinent solvent properties are tabulated below. An external supply and standard divider chain are used for the photo multiplier tube. The output from the photo multiplier goes to a current to voltage converter which drives a DC meter to indicate the average anode current. Do not exceed the maximum ratings for the tube: 1250V or 50 μA current (approximately full scale on the meter), whichever occurs first. After the meter, the DC portion of the signal is blocked, while the AC portion goes through a lowpass filter and then to the A/D converter. The filter starts to cut off the signal at about 2. kHz, and passes less than 1% of the input above 5.0 kHz. The output of the filter should be connected to the CH1 input on the acquisition input panel.

B. Data acquisition The raw data at each angle consists of several series of samples of I ( t ) over a time range 0! T. Each series is Fourier transformed and the average of the transforms is stored in a file for later analysis. The sampling procedure is repeated as necessary for other angles. These data are acquired with a PC, using an A/D converter board and a LabVIEW virtual instrument (VI). The remainder of this section explains how to configure the software and acquire these data. Several steps are needed to set up the apparatus: a. Position the arm in the desired position, turn off the room lights and adjust the HV to obtain an average current below half-scale on the IDC meter. b. Start the VI. Double-click the LabVIEW icon and then open the file NoiseSpectAve.vi in My Documents\LabVIEW Data. This should give a LabVIEW front panel. Go to View > Tools Palette and select the 'pointing finger' icon to get the control cursor. c. Set configuration parameters to match the hardware connections. Use the pull-downs or click in a box with the 'hand' cursor, type a value, and click ENTER or press the Enter key (Return won't work.). Typical values (with explanations) are: Physical Channel: Dev1/ai1 (analog input channel 1 on device 1) Minimum Value -5 (-5V expected minimum input) Maximum Value +5 (+5V expected maximum input) d. Set desired FFT and averaging parameters. Samples per channel: 2048 (must be power of 2) Sample rate (Hz): 10000 (to get Nyquist frequency > 5 kHz) Number of sweeps: 100 (desired number for averaging) Window Hann (general-purpose window) File output: OFF (until you want to save data)

e. Set FFT display parameters Display unit Vrms^2 (power is voltage squared) Log/Linear Scale Linear (but the semilog plot shows more detail) f. With the VI set for a small number of sweeps, click on Run. Windowed voltage vs time data will be displayed in the upper window for each sweep. After the specified number of sweeps the averaged power spectrum will be displayed in the lower window. Once the apparatus is adjusted you can obtain data at the chosen scattering angles by turning on the file output and running the VI. When the averaging is complete you will be prompted to enter a file name to save the data. (Pick distinctive names, so other students do not erase your files. If you intend to analyze with MATLAB on this machine, it is convenient to use the folder My Documents\MATLAB.)

Viscosity (centipoise) as a function of Optical index of refraction as a function temperature for water and methyl alcohol. of temperature for water and methyl alcohol

T(K) methanol water 280 0.7201 1. 285 0.6669 12409 290 0.6189 1. 295 0.5755 0. 300 0.5360 0. 305 0.5000 0. 310 0.4670 0.

T(K) methanol water 293.15 1.3284 1. 298.15 1.3265 1.

IV. Data Analysis

The goal is to decide whether or not the data support the theoretical calculation above. You may analyze your data on the lab PC, using the provided MATLAB code, or transfer the data to another machine of your choice. The required procedure is outlined below, followed by instructions for using the MATLAB routines.

A. General Requirements Agreement with the predicted Lorentzian form, Eq. 25, can be tested by fitting the averaged power spectrum at each angle. The power spectra extend up to the Nyquist frequency, typically 5 kHz, but we deliberately reduced signal frequencies above about 2.6 kHz to avoid aliasing. We must therefore eliminate the higher channels from the analysis, since they do not contain meaningful data. At the low-frequency end, the bottom few channels may be contaminated by "spill-over" from the large delta-function (zero-frequency) term and should also be removed from the fit. You should examine the fits to the remaining channels and check the!^2 values against appropriate tables to see if the data are indeed described by the Lorentzian form. You can also make a quantitative check, since the width parameter, in frequency units, is given by

2 Dq^2 2! = 16! Dn

2 "^2 sin 2 # 2

(26)

Evidently the fitted width should be linear in sin^2 "/2, with a slope that can be computed from the known properties of the sample(s). (Don't forget that the solvent viscosity depends on temperature.) By applying these checks to more than one sample we can say with some confidence that the theory does or does not apply in our situation.

B. Analysis using MATLAB Move your data files to My Documents\MATLAB, if they are not already there. Start the program by double-clicking the MATLAB icon. When the program prompt comes up, check that your files are in the current directory and then type lorentz0fit in the command window to execute the analysis m-file. It will ask for the data file name, plot your data set, and ask for the region of data to analyze. The selected data are fit to a Lorentzian function at zero frequency with a Marquardt algorithm, returning parameters and error estimates. You can repeat the analysis for another data set by typing lorentz0fit again, or leave MATLAB with File > Exit MATLAB.