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Professor Paul Russo Choppin 242
(225)5785729 paul.russo@chem.lsu.edu
Dynamic Light Scattering
Part I: Measurement of Diffusion of Polystyrene Latex
Part II: DLS of Dilute Polymer Solutions
Part III: Particle Size Distributions
Appendix: Data Fitting in DLS
Linear Fitting
Nonlinear Fitting
Laplace Inversion
Part I: Measurement of Diffusion of
Polystyrene Latex
In this exercise, we consider the determination of the size of your previously measured polystyrene latex samples by a completely different method compared to traditional light scattering. Its called either photon correlation spectroscopy or dynamic light scattering. DLS takes advantage of the high spatial coherence of laser light sources to produce instantaneous speckle patterns from any solution the laser illuminates. These speckle patterns move because molecules in the system move. In a dilute solution, the speed at which the speckle patterns move can tell us how fast individual molecules diffuse. The rates of motion easily exceed what can be followed by the best frame grabbers, so the whole speckle pattern is not measured, but only the intensity of a single speckle. Thanks to Einstein, we are able to convert the fluctuations in intensity at this one point into the size of the molecules that cause the scattering.
We shall begin by looking at a speckle pattern with our eyes. It will appear to move randomly. If we zero in on a small area in the speckle pattern (a coherence area) we shall see that the intensity in that area fluctuates. But the fluctuations are not really random! In fact, almost nothing is completely randomeverything that seems random is correlated (i.e., not random) on some sufficiently short time scale. For example, the Dow Jones industrial average produces a signal that fluctuates randomly about its mean value over some long enough time scale. (For simplicity, pretend that the mean value does not change; in reality stocks go up as new wealth is generated, not to mention inflation). But if the market is high at some time t = 0, then it is also likely to be large at some time t = e, provided that e is a very short time. Thus, if I put all my money into stocks today, I will probably not lose everything by tomorrow. On the other hand, at some time lag d that is very large there is no telling what the stock market will be like. It could be higher or lower; stock value becomes uncorrelated with the initial value after a long time. Investors and politicians would both love to know how long a bull market will last. Light scatterers are luckier; as you will see, our measurements are simple and precise.
Figure 1
Quasirandom phenomena are followed theoretically and experimentally with the aid of correlation functions. A correlation function is a mathematical construct designed to help determine how long a given signal stays correlated. For heavily damped phenomena such as diffusion, correlation functions decay approximately exponentially, with some characteristic time constant t, the "correlation time." In the foregoing, the time e < t while d > t.
Theory: The autocorrelation function measured will be of scattered light intensity:
G(2)(t) = <I(0)I(t)> = EMBED Equation.2 <1>
The (2) superscript indicates that G(2) is a secondorder autocorrelation functioni.e., one involving intensities, which are the squares of electric fields. The use of a capital G indicates that the data are not normalized. Later, we will see correlation functions like g(1) which represents a normalized electric field autocorrelation function and g(2) which is a second order function whose baseline is normalized. The notation of DLS is confusing; you eventually get used to it.
The integral is a fancy way for saying: record the intensity vs. time for an infinitely long period of time (i.e., very many correlation times). Take any and all pairs separated by the lag time interval t and "compare" them. The comparison is done by forming the products. When t is very short, one measures the average of the squares because I(t) ( I(t+t): G(2)(0)= . At very long t, one obtains the square of the average: G(2)(() = 2. Do you see the difference? Mathematically, is guaranteed to be larger than or equal to 2, so G(2) decreases with t. This process is repeated for many values of t until G(2) is known over a wide range of t. Actual digital correlators construct G(2) at many values of simultaneously, but it's easier for humans to imagine doing it for one value of t at a time. Operational details of how correlators work are discussed below. The curious may wish to read the various manuals, perhaps beginning with those for the LangleyFord DC64 or 1096 correlators. The operation of the more modern ALV correlator (and its only commercial competitor, the Brookhaven BI9000) is more complex.
After some time, the signal in the correlator is well approximated by:
G(2)(t) = B(1 + f(g(1)(t)(2) <2>
In this expression, B and f are experimental parameters that will be discussed later. The quantity of main importance is g(1)(t), the electric field autocorrelation function. In many cases, g(1)(t) is a simple exponential:
g(1)(t) = eGt <3>
where G is the decay rate (the inverse of the correlation time). For simple translational diffusion, the decay rate is:
G = t1 = q2Dm <4>
where Dm is the mutual diffusion coefficient. The scattering vector magnitude, q, is:
q = 4pn sin(q/2)/lo <5>
where lo is the laser wavelength in vacuo, n is the refractive index of the solution, and q is the scattering angle. Thus, the decay rate is lower (correlation time is longer) at low scattering angles than at high angles. You can also slow the fluctuations by shifting to a longer wavelength. The physical reason is that the molecules must diffuse farther to change the speckle pattern at low q. The distance scale of any scattering experiment is inversely proportional to q:
Scattering Distance Scale = 2p/q <6>
The raw data G(2)(t) is estimated at certain discrete values of t by a digital autocorrelator. There are two basic types of correlator, linear and logtime. The LangleyFord 1096 to be used in the present experiment exemplifies the classic linear design. A linear correlator measures the correlation function at discrete lag times according to t = iD where i = 1, 2, 3.....NCHAN and D is called variously the channel time, sample time, or sampling time.
The quantity f in Eq. 2 is an instrumental parameter (0 < f < 1) and B is a baseline, obtained in a variety of ways. One type of baseline is theoretical :
Bt = P(PO)/N <7>
where P = # of photocounts in acquisition time T (see Eq. 1) of the experiment and N is the total number of samples, or intervals into which time has been split: N = T/ D. The number of overflow counts (the number of times the correlator could not correctly read the intensity due to hardware limitations) is O, and should be zero (failing that, O should be many orders of magnitude less than P). In the LFI1096 correlator, these numbers are kept in various "housekeeping" channels: N comes from channel 273; P from channel 274 (which should be the same as channel 275). And, O is given in channel 276. All these channels can be directly read on the monitor screen. Just use the little toggle switch to set the cursor. In real experiments, all the values are sent instead to a computer by a serial line.
Putting it all together, we see that the correlation function has the form of an exponential decay on top of a baseline.
Figure 2
From this it is simple to see how to obtain the mutual diffusion coefficient, Dm, which differs in principle from the diffusion coefficient extrapolated to infinite dilution, Do. For the dilute latex suspension, without rigorous exclusion of salt, it will turn out that
Dm ( Do. Each student will make a measurement at a particular angle (different for each) and then do one manual analysis to obtain a diffusion coefficient and radius. You will just take about ten data points, using the cursor and screen display. Also, be sure to obtain the housekeeping channels so Bt can be computed. Make a note of the channel time D and temperature. You can check your answer by computer, using our "cumulants" algorithm CORAN if you wish. Once you know Do, the hydrodynamic radius is given by:
Rh = EMBED Equation.2 <8>
____________________________________________________________
Reporting:
1) If you are actually doing this HowTo (e.g., in one of our teaching laboratories) you should show the calculation you did to obtain the radius and diffusion. You must get the right answer, within several percent. As always, it is necessary to place some error estimate on the size. How you got your radius uncertainty should be explained. There should be a plot.
Diffusion Coefficient:__________ (cm2s1)
Radius:_____________________ (Angstroms)
2) What value of the coherence parameter, f, did you obtain?
f:__________________________
3) Do you need to know f accurately in order to obtain Rh?
4) What was the distance scale of your experiment?
Distance scale:______________________(cm)
Extra Credit 1: The experiment you just performed is called a "homodyne" mixing experiment. The reason for the fluctuations is that each illuminated particle emits light at a slightly different frequency. Why? Because the molecules move: DLS is really translational spectroscopy! When light of similar, but slightly different, frequencies is added, there occur "beat frequencies" that represent the frequency differences. In DLS, the beat frequencies are much, much lower than the frequency of light. They tend to be in the "audio" range: several kHz. You are actually measuring the linewidth of several thousand Hz, out of 1014  1015. DLS is the highestresolution spectroscopy known. Homodyne means "selfbeating": all the light that mixes together comes from the scattering cell. There is an older, lessoften used variant of DLS that mixes the scattered light with unscattered laser light. This is obviously called "heterodyne". It is still sometimes used for very difficult, unusual scatterers.
So how do you get the extra credit? Take the program you wrote earlier that generated a sine function. Make it generate two sine functions of slightly different frequencies, and add them. Now plot this. Do you see the beat frequencies?
Extra Credit 2: Devise or carefully describe a way to physically observe the beat frequencies in a wave mixing experiment.
Extra Credit 3: What does f depend on, experimentally speaking?
PART II: DYNAMIC LIGHT SCATTERING OF
DILUTE POLYMER SOLUTIONS
Objective: In the previous segment, you either read about or actually measured the size of a dilute colloidal particlea very easy assignment for dynamic light scattering. The present lab is harder in several ways:
Weak polymer solutions will be measured. We have to be concerned with the adequacy of the signal over solvent level.
In order to make measurements at all, the solutions must sometimes be more concentrated than one would wish. Then the results have to be extrapolated in to zero concentration in order to be meaningful.
Much greater care has to be paid to cleanliness.
Plan: We will use a commercial polystyrene (ideally, the same one selected during the simple GPC experiment) and try to make 4 samples with different concentrations, get their diffusion coefficients correctly, and extrapolate the diffusion coefficients to zero concentration. We will study the effect of dilution on the coherent scattering amplitude (see below). Later, we can use these same samples for conventional Zimm plot analysis and, perhaps, GPC/light scattering.
Preparation of Clean Samples
We cannot hide it any longer; the worst part of light scattering is preparing clean samples. With the latex samples we measured earlier, it was very easy: latex particles are almost as big as some dust particles and they can almost defend themselves from dust. In measuring most polymer solutions, you will not be so lucky!
Therefore, the first step is to prepare clean solvent! Yup.....just solvent, no polymer. Polymer analysts live by these words:
Measure Nothing First!
If you cannot do this well, forget the rest. Once you are measuring nothing well, measure something unimportant next, then measure the important perfectly (MNF ( MSUN ( MIP) But nevermind.....what will you put the clean solvent in? Clean Cells!
CLEAN CELLS
In our lab, we use three cellcleaning strategies:
Water Cleaning
Clean cell with soap, water, Chromerge or Alcoholic KOH
rinse
TEST every cell
dry
store in aluminum foil somewhere clean
Acetone Percolator Cleaning
Clean cell with soap & water
Rinse extensively
pop on Acetone Percolation device
Cells often dry quickly
TEST some cells, dry and hope for the best.
Cell surface modification (beyond the scope of this course)
WHATS CLEAN?
We define a clean sample as one in which no dust appears when the cell is inserted in the instrument and observed using the Argon ion laser at some fairly low angle, like 30o. This is a stringent test: the instruments optical magnification is about 40100(, and you can view a very large volume in the instrument, even though the measured volume is usually set to much less. Imperfections with sizes < 0.1 mm can be detected (not resolved, but detected). Many samples have been prepared that can be observed for many minutes at a time in this way. While such samples do exist (and if you don t believe it, some have been retained in a kind of dustless Hall of Fame) it is more common that one or two dusts will be observed eventually. Such samples often can be measured (maybe after centrifugation). If youre seeing something all the time, forget it and start over. Stay patient; some systems and cell types are harder to clean than other others, but I have never seen a system that cannot be cleanedwe always win! (Sometimes, winning means preparing just a few samples a week, but its worth it).
CLEAN SOLVENT
Distilling can help (but the glassware should be tested with clean water; for example, pour water from the collection flask into a clean plastic cell and see if it comes out OK)
Filtering can help (always check the chemical compatibility tables)
Centrifugation (the last resort)
As a first attempt, see if you can just filter a great grade of solvent into one of your clean cells. For most solvents, the pore size should be 0.1 mm. There isn t much dust smaller than 0.1 mm. We have occasionally found smaller filters useful, but the commonly available 0.02 mm Anotop filters must be tested carefully; not all these filters work. The ones that work at all work very well. We have large batch filtration devices for making lots of clean solvent if that helps. Ordinarily, we make it in small quantities that can be prepared on a syringe filter.
CLEAN POLYMER
Yup.....sometimes, it really helps to preclean the polymer. This is particularly true of highsurface area polymers (e.g., any powdery or fibrous polymer). Your average organic chemist (and is there any other kind?) likes to pad his or her yield with lots of crud from filter paper. Pelletized solid polymers (polyolefins, for example) are sometimes fairly clean. To clean your polymer:
Dissolve it in clean, tested solvent (at 12% typically)
Prepare a clean nonsolvent (test by holding it in the laser beam) in a clean beaker
Filter the polymer solution into the nonsolvent, using the smallest filter that does not plug or require excessive pressure.
Vacuum dry or freeze dry. Caution: some polymers misbehave on drying.
FILTER ADVICE.
Never force a solution rapidly through a small filter. Polymers can be degraded by shear forces in filters. This is a particularly true of large polymersmolecular weights above one million or perhaps even less for rigid, extended polymers.
It should almost never be necessary to filter a polymer solution more than once! If you have to do this, it means the collection vessel is not sufficiently clean or that you picked too large a filter size. You will often see frequent filterers in the literature: these are people who cannot clean their cells.
CLEAN SOLUTIONS
FINALLY! Take your clean polymer and add the clean solvent to obtain.....a dusty solution! Oh well, the bestlaid plans.... Usually, you will make your solutions in a volumetric or other large, screwtop glassware that will be imperfectly clean. With any luck (and you should not be in the light scattering game unless you are lucky) the dust will be of the easily removed, large variety.
Strategies vary for making a series of clean solutions, each with a different concentration.
Direct: If you have a large amount of polymer and many clean volumetrics, then just go ahead and make the solutions directly in the volumetrics. This is the most accurate wayespecially if you are sampling the output of a large factory without regard to preserving some precious polymer. However, each solution must be separately cleaned....and you have to test that the concentration of each solution is not affected by the cleaning.
SuperClean Stock: Make a stock solution in a volumetric and clean it to perfection. You will sometimes find that the stock solution has changed its concentration during cleaning. It is possible to compensate for this by measuring the concentration gravimetrically (or spectroscopically) after cleaning. With a clean stock solution available, you can make dilutionseither in clean volumetrics (correct) or directly in clean cells (may involve approximations).
Preparing dilutions in cells. The polyethylene tips on Pipetman type adjustable pipets are often quite clean (and they can be cleaned easily just in case). These adjustable pipets can simplify preparation of diluted polymer solutions from the precleaned stock. Sometimes the error of assuming volumetric addition is small. Other times, you may wish to actually obtain proper concentrations, as follows (example is for 20% cstock):
0) determine the stock solution concentration as c (g/mL) and w (g/g)
1) weigh a clean cell
2) deliver 0.2 mL of stock solution to a clean cell
3) weigh
4) add 0.8 mL of clean solvent
5) reweigh
6) compute c from the known weights
Functions and Settings on the LFI1096 Correlator
We discuss the LFI1096 because its simpler than the more modern ALV correlator or the correlator.com multicorrelator, which work by actually performing multiplications (I think) while the LFI1096 and similar BI9000AT perform multiple additions using an add command generator and shift register. Many of the same principles will apply to all instruments, so reading this section is important even if you wont use the LFI1096 (youd have to find it first!). The LFI1096 is a linear correlator (it has some weird modes, too, but we ignore these for now). As already discussed, its function is to approximate the integral:
G(2)(t) = = EMBED Equation.2 <9>
First, you must understand that the LFI is usually used as a digital correlator: it detects and counts discrete photon events. You connect the output of the photomultiplier tube (PMT) first to a preamplifier/discriminator (PAD), and then to the correlator:
Figure 3
So, Figure 1, showing a continuous variation of intensity, fails to represent the true situation. If we divide the time axis into discrete packets, all separated by a time interval D then we really measure a pulse train coming out of the PAD.
When building the average correlation function at lag time iD, products of the photocount samples of all times separated by iD are generated. The natural way to do this would seem to be to collect data for awhile, store it in a big array and then use a software algorithm to create the correlation function. This is called batch processing and no good correlator actually works this way (it is too inefficient; too much time devoted to computing and not enough to measuring). Instead, real time correlators are used, in which the data stream is pipelined down a shift register. The shift register contents move one channel to the right with each clock cycle, D. The contents of each shift register element can be added to memory every time a new pulse comes into a direct channel (the direct channel represents the present time ). In the figure below, we have added the direct channel and memory to the time diagram already shown, which now represents the shift register.
Also shown in the figure is the add command generator (the circle with the + sign in it). Whenever a pulse is detected in the direct channel, the each element in the shift register is added to the appropriate channel. For example, a pulse detected in the direct channel will cause the number two to be added to the memory associated with channels and 8. The memories of channels 1 and 6 would be increased by one. All the other channels do not change. If another pulse comes into the direct channel, the memories are again increased. Each time a new pulse arrives in the direct channel, the memories are increased. You can see this corresponds to multiplication of pulse counts, separated by times. For example, suppose 3 pulses arrive in the direct channel during some time period of duration D. The memory contents of channel 3 would be incremented by 3(2 = 6. This would be done by three separate additions. After a time D has expired, the data are clocked down the shift register, which now looks like this:
Now, with each new pulse in the direct channel, these data will be added to their respective memories. Thus, the products in the correlation function <I(0)I(t)> are built by successive additions n(0)n(t). The process is approximate because D is finite, because not all photons produce pulses (PMTs are not that efficient), because some pulses are false (arising from within the PMT and not having to do with light). For a detailed account of the approximations made in assembing a digital photocount correlation function in this way, see the textbook by Chu. A few points are all we need to stress.
It is clear that the ability of the direct channel to detect a rapid stream of pulses and the ability of the adders to add them to memory must be very high.
The shift operation of the register must be fast and free of losses.
Ideally, the shift register should hold very large numbers, but in practice it is limited. For example, the LFI1096 uses a 4bit shift register: numbers in each element can range from 0 to 15. Thus, if D were set to 0.1 s, the maximum average count rate would have to be much less than 15/0.1 s = 150 Hz.
Since some photomultiplier tubes produce dark count rates this high, prescaling may be required: only count every other pulse, every fourth pulse, every eighth pulse, etc.
The total time window is the number of channels times D. Extending the time window to handle five decades of time would require 100,000 channelsi.e., prohibitively expensive.
You must be sure to set D to capture the decaying process, and hope that you can capture all decaying processes in the available number of channels (which is 272 on the LFI1096).
These guidelines make for decent correlation functions with the 1096 or similar correlators.
Set the decay time first! Try to make sure you can capture all the processes. If the baseline is not laying down flat, use a different correlator (e.g., the ALV).
Try to keep the photocount rate to about 1/D. If the count rate greatly exceeds 1/D the natural fluctuations in the intensity will ensure that the count during some sample times will exceed 15i.e., you will overflow the shift register. This can cause great problems with analysis, so avoid it! If the coherence factor f is small, you can exceed this rate somewhat because the fluctuations in intensity will be small then.
We usually keep the average count rate under about 400,000/s, even if this is not yet 1/D.
There are quite a few other tips to LFI1096 operation, but these are best learned with practice. In the bad old days, you had to learn all the tricks, since linear correlators were commonly pushed beyond their limits. Now you would only use a linear correlator for fairly easy measurements. Logtime correlators such as the ALV5000 are now used for almost any difficult case. It is beyond the scope of this document to explain the mighty ALV (and its main competitor, the Brookhaven BI9000). However, a little information is provided, just so you can see that the limitations of the linear correlator can be dealt withusing radically different design.
Logtime correlators were suggested by the theoretical work of McWhirter and Pike, which will be discussed in greater detail later. These authors demonstrated that there was no advantage to having lots of linear spaced points if one wants to measure exponential decays (or multiexponential decays, as in polydisperse samples). A correlator that would work well could be constructed with just a few channels, whose lag times were exponentially spaced: ti = 2ti1. Alternately, you could use the same number of channels as a typical linear correlator (e.g., 256 or 272) and cover much greater time windows. About this same time, a number of workers discovered that wide time windows were really necessary to capture all the physics of many important processes (not usually in dilute solutions, however).
There were several attempts to stretch out the time window. The LFI1096 has an optional mode, called multitau, that divides system resources up so that three correlation functions can be measured simultaneously. The first uses a base time D between each channel; the second uses an extended time mD (m is called the divisor, never mind why) and the third uses m2D. The width of the shift registers expands so that, if the intensity is set so that overflows do not occur in the first correlation function, they also do not occur in the other two, even though the effective D is there very long. Operating in this mode, the window of the LFI1096 is 0  8192D. Thus, almost four decades of time can be spanned. Not bad, but not ALV either! Another patchedup attempt to extend the time window was made by Malvern, who made a correlator that looks kind of like this:
Note that memory only exists for some shift register locations. Thus, the data were delayed large amounts to generate the logtime spacing, but lots of data are being wasted in the shift registers. The performance features of the Brookhaven BI9000 suggest that it may operate this way, toobut I do not know for sure. There is not time to discuss the ultimate solution which is the ALV5000, developed by the late Klaus Schtzl, and discussed in the Wyn Brown book on dynamic light scattering. The ALV5000 has a completely different architecture, using (I think) multiple 8bit processors and the CPU on a host IBMPC computer to actually multiply the lagged intensities together very rapidly instead of add them repeatedly. Data are stored in memories, and precision varies from 8bit to 16bit, depending on the lag time. There are a number of other tricks to making the data come out very accurate, but the main point of the ALV is the same as other new correlators: it achieves very wide sample times with little hardware. Also, you dont have to aim the available channels at the actively decaying part of the correlation function. If it moves, the ALV will likely capture it. There is no decay time increment to set.
Acquiring Multiple Data Sets
Assuming you have demonstrated the ability to compute diffusion coefficients in the previous lab on latex spheres, you can use our computer programs, LFI232 and CORAN. LFI232 gathers multiple runs; CORAN allows you to analyze each run quickly, and you can throw out any bad runs. The remaining runs are summed together to obtain a quiet run. In any such summing procedure, there are some practical restrictions and precautions.
1) The individual short runs must have an acquisition time Tshort which is at least 1000 times longer (preferably much more) than the longest correlation time. This enables each measured short correlation function to be a reliable estimator of the true correlation function. Always remember: a correlator only estimates the true correlation function. Not only that, but individual runs can differ some, for myriad reasons.
2) It is traditional with the LFI1096 to look carefully at the baseline channels, which should be flat. Use the Display Mode button and menu to blow up the scale and check.
3) It is a very good idea to test whether the measured coherence parameter f meets with expectations. On the LFI1096, measured f values are easily computed by reading initial and baseline values with the toggle switch cursor. But what is the expected value?
Let fmax = the f value you would measure with a very strongly scattering sample that exhibits no longterm decay anomalies. Examples are latex, microemulsion, and silica sphere solutions. The sample should also exhibit no very fast, shorttime decays. The value of fmax depends on the aperture and pinhole settings, laser wavelength, beam focusing, photomultiplier dark count rate and, to lesser extent, scattering angle. The expected f value is reduced from fmax due to incoherent (on the time scale of the autocorrelator) scattering from the solvent. The figure below shematically shows the correlation functions of latex spheres and pure solvent (e.g., water or toluene). A linear yscale and log xscale results in sigmoidally shaped plots for a normal exponential decay. The plot is drawn for high f values (i.e., the lowtime yintercept is almost twice the baseline).
The scattering from rapidly diffusing solvent molecules only remains correlated for very short times, inaccessible to correlators. The limit for the LFI1096 is 107 s, as shown; a stock ALV5000 is limited to 2 ( 107 s. A soupedup ALV5000 and the new Brookhaven BI9000 can reach about 1 x 108 s. However, the solvent decay time lies still further below this for most normal solvents. Thus, the expected f value for a solvent is usually zero.
The expected f value of weakly scattering polymer solutions will lie in between zero and fmax. It can be computed from:
fexpected = fmax EMBED Equation.2 <10>
where Ap is the scattering amplitude associated with the polymer and As is the scattering associated with solvent (i.e., the solution scattering is Atotal = As + Ap). Thus, if the solution scattering is twice the solvent scattering, expect f to be decreased to 25% of its maximum value.
What if f is actually less than fexpected? This is very valuable information! It can mean two things:
1) The experiment is not being conducted in the homodyne modei.e., there is stray light or a deliberately added local oscillator to force the heterodyne condition.
2) The polymer dynamics are too fast to capture with the correlator.
Conversely, if the measured f value is equal to the expected, it means that the correlation function has been collected correctly in the homodyne limit and that all the decay modes present have been captured.
4) The total acquisition time T should be something like 106  109t. It could be even longer in cases where very quiet data are required for the initial part of the decay (where the noise is determined by photon starvationi.e., there are few photons per sample time for very short times).
Data Analysis
This section is sketchy. More information can be found in the Appendix. The CORAN program we will use performs a socalled cumulants analysis to obtain the average decay rate EMBED Equation.2 = q2Dm. The diffusion coefficient is given by:
Dm = Do(1 + kDc) <11>
where Do is the zeroconcentration extrapolated value, and kD is concentration dependence, which contains thermodynamic and frictional parameters. Sometimes this expression is written with more detail:
D = M(1 v2c)( EMBED Equation.2 <12>
where ((p/(c)T is the osmotic susceptibility, NA is Avogadro s number, M is the molecular weight, v2 is partial specific volume of the polymer, fo is the friction coefficient for the polymer, extrapolated to zero concentration. For a derivation, see Yamakawa: "Modern Theory of Polymer Solutions". For still another form of this equation, see Eq. 17 of Varma et al., which is derived using an expression for ((p/(c)T which you should look up in, for example, Yamakawa's book or Tanford's.
In any case, our goal here is not to worry about the dependence of diffusion with concentration; this is a research project. Our goal is to verify that the effect really exists and then to "extrapolate it away," which is the standard approach in polymer analysis. By measuruing the diffusion coefficient at several concentrations, the value extrapolated to zero concentration (where there is no thermodynamic interaction or interference from neighboring polymer molecules in solution) contains just information about the molecule. Specifically, the zeroconcentration diffusion coefficient is related to molecular weight by:
Do = aMb
where a and b reflect the dimensions and scaling properties of the polymer, similar to the MarkHouwink intrinsic viscosity parameters. It should be noted that these parameters are not very welldetermined in the literature at the low molecular weights of present concern.
Procedure
As always, polymer solutions should be prepared by adding a SMALL amount of solvent, allowing the polymer to dissolve, THEN bringing the solution up to volume. Failure to do so causes real problems with polymer solutions. For example, a fiveml solution is prepared by adding 12 ml of solvent to the volumetric, allowing dissolution to take place (add stir bar if needed, but don't add one to a highly viscous, tacky solution), then raising the volume up to the mark.
See if you can succeed without precleaning the polymer. When you add solvent, use a filter (see above) and lets hope these filter out OK.
Be sure to prepare a clean solvent (toluene) in addition to the several solutions of differing concentrations.
Measure the intensity of your solutions and of clean toluene. Measure correlation functions for each (dont worry, Ill help you).
Measure a latex so you know fmax.
Obtain the diffusion coefficients as a function of concentration. See if the expected f values are actually obtained.
What to include in your report
CORAN outputs
GPLOTs (nice plotswell show you)
analysis of fmeasured vs fexpected.
Plot of Dm vs. c, identifying Do ( and kd (
Part III: Measuring Size Distributions by DLS
It is not always realized, especially by chemists, that a really important thing about macromolecules is that they are big. Colloids are also big, but they are not necessarily molecular. In this lab, we will use dynamic light scattering to measure the size distribution of a common colloid: milk. Materials safety data sheets are not required, except if you suffer milk allergy and plan to drink the supplies.
What we are going to do is measure an autocorrelation function, as we already have for latex spheres and for polystyrene. Whereas latex particles are very uniform in size, yielding a single exponential decay, and commercial polystyrene is reasonably uniform, yielding a correlation function that could be handled by cumulants, colloidal milk contains broadly polydisperse particles of fat bound up with proteins and other molecules. The correlation functions will have pronounced nonexpnentiality. Two Laplace inversion routines will be used to sort out the decay spectrum, and then you will be asked to make a conversion to a size spectrum on your own, making the appropriate corrections for particle form factor, assuming a spherical particle shape. Your answer will be tested against the crude but effective software that we use in our own research.
You are reminded that the main quantity of interest in DLS is the electric field autocorrelation function, g(1)(t) which for a polydisperse sample consists of a sum of exponential decays:
g(1)(t) = EMBED Equation.2 <13>
An amplitude Ai is proportional to molecular weight and concentration of species i, modified by the particle form factor for sufficiently large particles:
Ai ~ ciMiP(qRg,i) <14>
At infinite dilution, a decay rate Gi is related to the diffusion coefficient (hence, hydrodynamic radius) of that particle and the scattering vector:
Gi = q2Do = EMBED Equation.2 <15>
where Stokes' law was used in the last equation to relate Do and hydrodynamic radius Rh through the viscosity ho. In the present experiment, we will assume we are at infinite dilution.
The particle form factor depends on shape; you must know it (or assume it) and then refer to tables (for example, in Kratochvil chapter in the famous book by Huglin). Also, in order to obtain P(qRg,i) one must convert from Rh,i to Rg.i. This conversion also requires the assumption of a shape. For solid, spherical particles (Rh = R) of uniform density, we can write:
Rg.i = XRh,i <16>
where
X = EMBED Equation.2 <17>
We will prepare a milk suspension, reasonably free of dust, and measure it, probably at q = 90o (depending somewhat on scattering angle). We will place a file on a disk (that you must supply) containing the correlator output in ASCII form. One of the several data columns in the disk file contains the lag time, t. Another column contains the square of g(1)(t)  i.e., (g(1)(t)(2. We will tell you which columns.
WHAT YOU MUST DO
1. Do It Yourself Data Fitting
You should try to obtain the distribution c vs. R yourself. One way to do this is to write a short program that first reads and then plots the experimental g(1)(t) data on screen. Add a little routine that generates a fitted data set according to Eq. 1, where you supply the values Ai and Gi manually. You could plot this fitted line in a different color easily enough (see programming examples from beginning of course). In a Visual BASIC implementation, it would be simple enough to use menubars to adjust the A s and G s. It is also quite possible to use Origin or a spreadsheet to good advantage for these same purposes. In Origin, look under the Fit/ParametersSimulate option. For daytoday use, this is slow compared to a purposebuilt program, but its OK for what you are doing. I encourage everyone to at least try to write their own program. "Canned" graphing packages are great until you have to use them again and again. Then their general purpose baggage becomes too much. An intermediate solution is using the scripting option of Origin, but I don t actually know too much about that!
The best approach is to fit A vs. G first. There is no way to tell a priori how many exponentials this will take. Make a good guess at G by inverting the lag time at which the correlation function is actively decaying and try to fit the function with just one exponential. Semilog plots (ln(g(1)(t) vs. t) can be very helpful. Then keep adding exponential decay terms until the randomness of the residuals no longer improves. More than 3 or 4 exponentials will probably not be required.
After you are satisfied with the fit quality, you can worry about converting the G s to Rh s and, subsequently, Rg s. And then you can convert the A s to c s. Well, actually you cannot directly obtain the c s: you lack the constant of proportionality in Eq. 14. So, what you will do is divide through the amplitude Ai by MiP(qRg,i):
ci,pseudo = Ai/MiP(qRg,i) <18>
where ci,pseudo is proportional to the concentration. But you must do something to estimate Mi from Rh. To really do this right, you would need the density. But on the assumption that the density of all particles is the same, you could just replace Mi in Eq. 18 with EMBED Equation.2 .
However you do it, you should discover that fitting data this way is a very dicey business! It will be interesting to see how close you can all come to each other. A student will be appointed to gather the results and make this comparison.
2. Use our Data Fitting
We have several programs that reliably give fits to polydisperse data. After you have successfully fit the data and demonstrated your own method to me, you can use our software to see how close you came to the right answer. The following Appendix tells a little about how our software works, and it gives a little general DLS advice.
Appendix: Data Fitting in DLS
Linear Fitting
The simplest cases are latexes or other nearly monodisperse scatterers. These are easily fitted using the usual linear fitting routines found in the classic book, Data Reduction in the Physical Sciences, by Bevington, or in the more modern but not necessarily better book entitled Numerical Recipes by Press, Flannery and Vetterling. Our programs use Fortran code from Bevington, converted into QuickBASIC or PASCAL in some cases.
Equations 2 and 3 can be combined to read:
G(2)(t) = B(1 + f e2Gt) <19>
This is linearized to:
y = ln(G(2)(t) B) = ln(Bf)  2Gt <20>
Thus, a plot of y vs. t has slope 2G, from which Dm can be extracted, since G = q2Dm.
The figure shows the ideal, linear behavior as a straight line. Polydisperse samples look like the curved line instead. The initial slope is called the first cumulant and given the symbol EMBED Equation.2 because it represents the average decay rate (You should try to convince yourself that EMBED Equation.2 really does represent the average decay rate). In order to get the first cumulant, a polynomial fit is performed on the y vs. t curve. This is called cumulants analysis; the order of the cumulants analysis means how many terms: a first order cumulant analysis just fits a straight line, a second order analysis a quadratic, and the third order analysis a cubic, etc. In practice in our laboratory, we commonly take the first cumulant from third cumulants analysis (confusing, isnt it?). If the third cumulant analysis does not well represent the data, it is time for something more sophisticated (see below).
Cumulants were introduced by Koppel, who showed that EMBED Equation.2 was proportional to the zaverage of the diffusion coefficient:
EMBED Equation.2 = q2Dz <21>
where
Dz = EMBED Equation.2 <22>
I will leave it to you as an exercise to figure out which average of hydrodynamic radius is obtained! Note: it is not the simple zaverage as is often claimed in the literature by people who dont know better. People write Koppels expression differently; I write it like this:
ln(g(1)(t)) =  EMBED Equation.2 t + EMBED Equation.3 m2t2 + ..... <23>
The term m2 is called the second cumulant. For a perfect single exponential decay, it would be near zero. Like the first cumulant, the value one obtains for the second cumulant of nonexponential recoveries varies a little bit with which order of fit you are using. (One of the disappointments of cumulants fitting is how sensitive this can be for weird decays). A measure of the polydispersity of the distribution can be given in terms of the unitless quotient m2/ EMBED Equation.2 2. DLS people sometimes call this the polydispersity parameter or normalized variance. In most of our programs, it is called POLYD
POLYD = m2/ EMBED Equation.2 2 <24>
POLYD is hard to measure well, involving as it does a ratio of two quantities that themselves depend a bit on the order of fit. As a rule of thumb, if POLYD > 0.3 it is time to consider another approach. Cumulants analysis should be reserved for data screening (e.g., the program CORAN) and for nearly single exponential decays.
We have been doing cumulants analysis for a very long time, but we still continue to learn. The arrival of the ALV correlator, with its very wide ranges of lag times t, has obliged us to be more careful about how data are weighted for noise while doing cumulants. Also, with any cumulants package, it is essential to delete some channels near the tail: some of these will go negative when the baseline is subtracted, and log(negative) operations are not well liked by computers.
Nonlinear Fitting
In cumulants analysis, the parameters of interest (m2, EMBED Equation.2 ) appear as linear coefficients of the independent parameter, t, in eq. <23>. Cumulants fitting is like a small perturbation applied after a huge linearization operation. More generally, we might try to fit G(2) or g(1) directly. If g(1) = EMBED Equation.2 then we could look for Ai and Gi by trial and error. The Marquardt algorithm (see Bevington) makes this process as rational as possible. In this algorithm, one makes initial guesses at the A s and G s. The program looks in a semiintelligent fashion for better parametersi.e., ones that reduce (but perhaps do not really minimize) the difference between fitted and actual data. The parameter c2 is monitored to assess the progress:
c2 = EMBED Equation.2 <25>
The function yfit is a multiple exponential whose amplitudes and decay rates are adjusted. It is evaluated at precisely the same ti where the experimental data were evaluated. The symbol n represents the number of degrees of freedom, approximately the same as the number of data points, N. The statistical weight wi is the inverse, squared uncertainty of each data point: wi = si2. The meaning of c2 is this: when it is unity, the errors of fitting are comparable to the uncertainty in the measured data. The data have been fit to within their reliability, using the model function yfit. The simplest trial function yfit to produce this desired result is preferred.
Be careful interpreting c2! A lot of students think that high c2 means something is wrong with the data. This is one possibility, but not the only one! The c2 parameter is the result of data quality, data noise and the adequacy of the fitting function. Suppose you have a genuinely nonexponential decay, but are fitting with just a single exponential term. Then, a high c2 value doesn t necessarily mean anything is wrong with the data. In fact, better data (lower s) will increase c2. A legitimate use of c2 is when comparing multiple runs of similar quality. If a particular run has a c2 much higher than the others, then that run may be defective. If all the multiple runs have similar c2 values, but these values are high, it possibly means that a better fitting function must be selected.
One should never forget that nonlinear fitting is prone to give false minima. There is actually a hyperspace where c2 could be plotted against many parameters (for example, two A s and two G s in a twoexponential fit). There is some particular combination of A s and G s that really produce a minimum c2but there could be lots of local minima. To avoid getting stuck in a local minimum, the initial guesses are varied and one sees if the Marquardt algorithm will steadfastly return the same best values. If it does, then it is assumed that these fitted parameters really do represent the data. This is quite a different situation than linear fitting, where the best parameters of fit are determined analytically!
The raw data G(2) may have lots of decaying exponential terms, in general. For example, if g(1) has two terms (g(1) = EMBED Equation.2 + EMBED Equation.2 ) then the active part of G(2) (which is just (g(1)(2) must contain three exponentially decaying terms. They have amplitudes A12, 2A1A2 and A22 with decay rates, respectively, of 2G1, G1 + G2 and 2G2. However, the three decay rates are not independent. If g(1) contains just two exponentials, and you fit G(2) to three exponentials, then the decay rates of those three exponentials should be tied together. Sometimes, they will not be: this either indicates that g(1) does not contain just two exponentials, or that some error in measurement has occurred. A good test for two exponentials (and no more) is to fit g(1) to two exponentials and G(2) to three exponentials. A consistent set of G1 and G2 should result.
Our homebrew multiexponential, nonlinear fitting software is called MARLIN (for the ALV, MARLINA). Nevermind why. It s a slight adaptation of the routine CURFIT found in Bevington. Since it only fits sums of exponentials, one must specify a baseline if one wishes to fit g(1). The program easily handles fitting G(2) but, except possibly in the case of a bimodal sample, the decay rates from G(2) are difficult to interpret.
Laplace Inversions
In many cases, a nearcontinuum of scatterers is present. For example, in a polycondensate with Mw = 200,000, the individual species may differ in molecular weight by, say, 100. Instead of writing g(1)(t) = EMBED Equation.2 we can write, to a good approximation,
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The big question is: knowing g(1)(t), can we obtain A(G)? The two functions are Laplace transform pairs, and the process of inverting g(1)(t) to get A(G) is called Laplace transformation. The idea is similar to Fourier transformation in that functions in reciprocal variables are involved, but the actual process is not like Fourier transformation because uniquely best answers are very difficult to obtain. Fourier transforms are easily computed using the famous Fast Fourier Transform algorithm. Provided that sampling theorem considerations were respected when the measured data were taken, the FFT is not very sensitive to noise and consistent results are obtained. The Laplace transform process fares worse: noise, even a very small amount of noise, really leads to problems. Also, unlike Fourier transformations, no single, universallyaccepted, fast, efficient algorithm exists for Laplace inversion. Indeed, there are still a few DLS researchers who think Laplace inversion is a black art and even statistically unsound! This view is probably extreme, but Laplace inversions really should be scrutinized and doublechecked.
Despite the frequent existence of multiple exponential decays in a variety of natural and measurement processes, it was not until the late 1970s that the situation was understood. The paper by Ostrowski et al. contains a nice discussion of the original work of McWhirter and Pike. The essential discovery of these workers can be summarized:
If the data in tspace (the correlation function we can measure) contain any noise, then the information available in G space (i.e., the function A(G) that we desire) is limited to low resolution: fine details of A(G) will be extremely difficult to obtain.
Remember, A(G) is really c(M) because A converts to c and G converts to M, with a series of approximations. The implication is that, if the true distribution looks like this:
we might be very lucky to measure something instead like this:
If you think in Fourier series terms, the sharp little blip in the highM side of the true distribution requires some high frequency components. In this context, frequency means that lots of M values would have to be included in our sampling of the distribution: we would have to know the difference between c(M) and c(M+dM) where dM is small. Although the basis set of Laplace inversion is not simple sine or cosine terms, the frequency analogy is still apt: the highfrequency terms required to correctly describe the blip are simply not available if g(1)(t) is noisy and it doesn t have to be that noisy!
Pike and McWhirter liken the information retrieval process to image formation by a lens. Bob Detenbeck (U. Vermont) makes a wonderful analogy to audio reproduction. Trying to determine A(G) from g(1)(t) is like trying to decide the type of violin being played by a gifted soloist from an AM radio broadcast: the highharmonic overtones that you require to distinguish one violin from another are simply not present at any level above the noise (static) of the broadcast. The reason is that AM radio has a severely limited bandwidth: frequencies above, say, 5000 Hz are not reproduced. However, noise is presentincluding some components above 5000 Hz. The correlation function is like a bad radio or amplifier: it just cannot transmit the highfrequency components with any significant amplitude. They get buried in the noise. If you try to guess the highfrequency components (i.e., finer details of the distribution) you will often get the wrong answerbecause your guessing is based on the noise as much as the real signal. Similarly, if you try to guess whether Yitzakh Perlman is playing his Stradivarius or some other violin, while listening to an AM radio broadcast, you are likely to get the wrong answer. You may be able to discern that a violin, and not a viola, is being played. In DLS/Laplace transform, you will be able to tell that molecules are big or little. Perhaps more sometimes. Dont expect much and you wont be disappointed.
In Fourier analysis, one obtains the amplitudes of sine or cosine functions that, added together, give the waveform of interest. The sine functions have frequencies like g, 2g, 3g, etc.i.e., the frequency of the functions to be superposed linearly increases...but it is discrete. The essential bit of information for the PikeMcWhirter analysis of the Laplace inversion operation is that there is just no point seeking such detailed information. It is still possible to represent the distribution using a discrete set of (exponential) functions. But the decay rates of the exponential functions whose amplitudes we seek should be spaced farther apart. Instead of looking for A(G), A(2G), A(3G), we should space the decay rates evenly in a logarithmic space. This can be expressed:
Gi+1 = Giep/wmax <27>
or, equivalently:
ln Gi+1 = ln Gi + p/ wmax <28>
The parameter wmax is set empirically according to the noise level: for less noise, use a higher wmax so that the distance, in log space, decreases. This corresponds to more resolution. For noisier data, do not attempt such resolutions. Decrease wmax so that the distance between functions (they are called grid points as you will see below) increases.
The suggestion of McWhirterPike is to sample the true distribution, using a discrete number of grid points (exponentially decaying functions) whose decay rates are exponentially related. They called this exponential sampling. Suppose the true distribution looks like this:
How would you sample this function? First, you would have to define the minimum and maximum ranges over which to seek solutions. Call these GAMMIN and GAMMAX; they set the RANGE of the inversion process. Next, you would select wmax and select some grid points that fall inbetween GAMMAX and GAMMIN. The number will usually be something like five.
The vertical arrows show the evenly spaced grid points. The next step is to determine how long the arrows should be, according to the only information we have about the true distribution, which comes in the form of the imperfectly measured correlation function. This is done by taking the function EMBED Equation.2 where the G s are now specified by Eq. 27 and fitting it to the data by finding the best A s. Since the decay rates are fixed, this is a linear fitting problem. There would be no point trying to float five decay rates and five amplitudes; that should just about fit any decaying function and might violate the McWhirterPike guideline, since floating decay rates could float very close to each other.
One thing may bother you. Five little grid points do not look very much like a continuous distribution. There are two solutions to this. The first is to use an interpolation formula (see Ostrowski paper). The second is to shift the grid points. In practice, we use the latter option. The whole set of grid points is shifted by a distance p/5wmax (there is nothing magical about the number 5). The new grid is shown below, and the lengths of the original grid point arrows have been extended according to the best fit obtained. Note that the arrow lengths do not exactly match the true distributionthats the result of noise. But remember, in practice, you do not know the true distribution!
Now the process is repeated. The new grid point arrows are lengthened, according to the best fit for that grid. Then there occurs another shift by p/5wmax, another linear best fit, etc. Eventually, the true distribution is approximated.
But how do you select wmax and thereby control the number of grid points? And, for that matter, how do you set GAMMAX and GAMMIN? The answer is to set GAMMAX and GAMMIN very liberally at firstinclude decay rates that are too fast and too slow. Set wmax to some low value, like 3. Try to fit the data. Probably, the best fit will contain some negative amplitudes. This is physically unrealistic! Molecules cannot have negative molecular weights or concentrations, so we enforce a nonnegativity constraint. To do so, delete those grid points from the fit and repeat (some exponential sampling algorithms do a bit better and rearrange the grid). In this way, the physically meaningful RANGE will be identified. Then try to add as many grid points as possible by raising wmax. If wmax is too high, your fit will again respond to the noise, not the real g(1) signal, and you will have to back off and/or decrease your range.
If this process sounds laborious, it is! We still use it, however, to scope out an inversion in preparation for automated Laplace inversion routines. The most important of these is the Fortran program CONTIN, written by S. Provencher, an American emigre to Germany. Provenchers program is a standard for the DLS crowd, and it is used elsewhere too (e.g., fluorescence and DOSY NMR communities). It does not rely on sequential stepping as does exponential sampling and, therefore, is capable of returning narrower distributions. Actually, CONTIN generates up to 12 different answers and then automatically chooses the one it thinks is best. To CONTIN, best means the least detailed distribution that is consistent with the data. It makes this choice based on statistical estimates (pretty vague statement, eh?) Some more detailed distributions will be obtained and some less detailed ones too. The user should always inspect all of them. Our software makes this easy to do. CONTIN competitors include exponential sampling (but there is no standard, and everyone writes their own program; ours is quite good). Other CONTIN competitors are based on the Maximum Entropy approach; these have received mixed reviews.
Another thing should be mentioned. As usually used, CONTIN and also our own version of exponential sampling, called EXSAMP, do not just minimize c2. Rather, they minimize a modified c2 where a term has been added to penalize fits where adjacent grid points produce dramatically different A values. Thus, unrealistically sharp variations in amplitudes are reduced. This is called enforcing parsimony. This is discussed in two articles from our lab, and in many other places.
As already mentioned, Laplace inversion has its detractors. Most people who look at the problem are amazed that Laplace inversion cannot be done with excellent resolution. For example, 32 grid points is a commonly used CONTIN configuration. The number of data points collected is usually much moreperhaps 272. To the uninitiated, it may seem that 32 parameters might be fitted to 272 points without so much trouble. Well, it aint so. On the other side of the spectrum are experienced dynamic light scatterers who think that CONTIN, exponential sampling, etc. are all too much detail, and that one should stop with cumulants, double or perhaps triple exponentials, or stretched exponentials (we havent discussed this option). These people point out that 32 functions gives way too many adjustable parameters, that it defies logic and statistics to use so many, etc. I think that position is extreme, too. CONTIN and exponential samp2X8X:XzXXXXNYTYnYpYYY]Z`Ze^k^^``<`>```a6a8aaab bAbCbDbEbIbKbbbbbc!cddddeegggghhnnrrrrrrrrrrrrrrrrrhk:H*OJQJ hk:H*jhk:CJUmHnHuhk:6OJQJhk:OJQJ hk:H*hk:hk:6OJQJ hk:6HGbHbIbLbMbNbbbbbbbbbbRfTfllrrs0s2srsts#v$v9w$a$rrrss2s8s:s@sFsLsNsRsVsXsZs^sdshspssss8t:t@thujupuLvSvvvvwww:w;wBwEwVxXx^xPyVy`ylyvyxyzyyyyzz{{{{ƾj>3
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hk:CJH*=9ww?w@wAwBwCwDwEwRyTyVybydyfyhyjylyxyyyyy~~hij}}ojtuvy{Gb7CQJLNMNdeÿjPM
hUVjhUhhWghWghWg5hWgh5U hk:H*jhk:CJUmHnHu hk:H*hk:OJQJ hk:6hk:Ajuwxy8:ؐڐNOdf@Amnling algorithms attempt to take advantage of the solid theoretical work of McWhirterPike, which defines about how much you can expect (i.e., not much!). The programs attempt to construct a logical, repeatable method to take advantage of what is available. With parsimony, these functions do not really overfit the data as badly as it may seem; the amplitudes are tied together because the objective is to minimize the modified c2. As usual, the intermediate position is best: use Laplace inversion programs, but use them with great caution and respect for the fact that resolution is inherently poor.
Some guidelines (from experience)
Exponential decays cannot be resolved unless the two decay rates differ by more than a factor of about 2.
Lowamplitude peaks are especially suspect.
Always confirm by applying two inversion routines.
Always apply multiple exponential fits in addition to Laplace inversion.
Dont feed these programs bad data!
Always investigate the effects of modest baseline changes on the distribution.
Milk: A Simple, Practical Example
In June 2009, an old friend asked me to look into the distribution of milk. Here are some results based on scattering at a single angle (always a bad idea) of 30 degrees from diluted milk in 3 grades. This is an easy experiment to do; comparisons to the Malvern zetasizer, which measures at 173 degrees, showed somewhat smaller sizes. An advantage of the Malvern, though, is that you do not have to dilute the milk at all. Its clear that DLS can detect these differences; how accurate these distributions are is another matter entirely.
EMBED Origin50.Graph
REFERENCES
B. K. Varma, Y. Fujita, M. Takahashi and T. Nose, J. Polym. Sci., Polym. Phys. Ed., 1781 (1984).
B. Chu, Laser Light Scattering, 2nd Edition. Academic Press (1991).
C.M. Kok and A. Rudin, Makromol. Chem., Rapid Commun., 2, 655 (1981).
Ostrowski, N., Sornette, D., Parker, P., and Pike, E.R. Exponential Sampling Method for Light Scattering Polydispersity Analysis. Optica Acta 28:1059, 1981.
Koppel, D.E. Analysis of Macromolecular Polydispersity in Intensity Correlation Spectroscopy: The Method of Cumulants. J.Chem.Phys. 57:48144820, 1972.
Schatzl, K. in Dynamic Light Scattering: the Method and Some Applications, Brown, W., ed. Cambridge Press: New York, 1993. Ch. XXX
Huglin, M. B., "Light Scattering from Polymer Solutions" (a preDLS book on LS).
The textbook, "Numerical Recipes" by Press et al.
Russo, P.S., Saunders, M.J., DeLong, L.M., Kuehl, S.K., Langley, K.H., and Detenbeck, R.W. ZeroAngle Depolarized Scattering of a Colloidal Polymer. Anal.Chim.Acta 189:69, 1986.
Guo, Y., Langley, K.H., and Karasz, F.E. Restricted Diffusion of Highly Dense StarburstDendritic Poly(amidoamine) in Porous Glass. Macromolecules 25:49024904, 1992.
Actually, laser light is not required. Spatial coherence can be achieved by other means.
Recently, some relatively slow processes have been followed with frame grabbers, many speckles at a time.
The approximation is that the scattering is homodyne and a random Gaussian process.
If multiple relaxation times are present, use the longest one. In such cases, the ALV5000 correlator is superior to the LFI1096. But the shortrun acquisition time must still be several orders of magnitude than the longest decay time.
A possible cure if one is not really at low enough concentration would be to make a kD correction, assuming that kD does not depend strongly on molecular weight and that the overall polymer concentration can be used.
http://russo.chem.lsu.edu/howto/HowTo.html..DLS.doc
Paul Russo June 5, 2009
PAGE 31
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