## Characterization of polymers by light-scattering

### Introduction

Historically, the first scientist to discuss the scattering of light from small particles (in any scientific sense) was Leonardo da Vinci, who correctly deduced that the blue of the daytime sky was due to the scattering of light from "minute and insensible atoms on which the solar rays fall"and to a blackness behind the atmosphere, rather than any innate color. Only 350 years later, Leroy Tyndall published experiments on the scattering of light from suspensions in transparent fluids. And shortly after that, Lord Rayleigh used Maxwell's newly constructed wave theory of light to analyze theoretically the scattering of light from suspended particles.

### Rayleigh Scattering small molecules  ### size<<l/20

Rayleigh's result for the scattering of light from small particles is shown above. An incident beam of irradiance I0 (=power/unit area carried by the electromagnetic wave) and wavelength l is incident upon a small particle of polarizability a. The polarizability of a particle may be loosely considered as the ease with which the incident electric fields deform its constituent charged particles from one another (although we will revisit the notion of polarizability). The irradiance of the energy scattered by such a polarizable particle is given by Iscatt, which is seen to vary as an inverse square of the distance from the scattering object, as any decent spherical wave should. It is not a uniform spherical wave, however, as is indicated by the sin2F term, which shows that the scattered intensity is maximum in all directions perpendicular to the incident electric field and zero in the two directions parallel to the incident electric field. Although we will not delve into it, this behavior is responsible for the peculiar reflective properties of light incident on a transparent medium at the Brewster angle (the only angle I know of that is patented). Note also that the amount of light scattered varies inversely as the fourth power of the light wavelength, showing that blue light (of wavelength ~400 nm) is scattered about 5 times more effectively as red light (of wavelength ~600 nm), giving Rayleigh the credit for quantitatively explaining the blue sky and the red sunrise/sunset. (Also buried in here is the explanation for the polarization of light scattered by the atmosphere - if you wear polarizing sunglasses, look up at the sky, about 90º from the sun, and rotate your head [put each ear on your shoulder and repeat] and note the appearance of the sky).

For solution work, it is useful to define a so-called Rayleigh ratio, as is done at the bottom of the figure above. (Since for solutions, Iscatt is defined as the scattered light per unit volume of solution, the Rayleigh ratio has units of reciprocal distance). The constant K is given by

2p2n02(n/c)2 / NA l4 and includes experimental constants like the wavelength of light (l) doing the scattering, the refractive index of the solvent (n0) and the change in index upon addition of a suspension to the solvent (n/c, the refractive index increment) (NA is Avogadro's number). Note that what is measured is the extra scattering due to the addition of polymer to solvent. That there should be scattering from a transparent liquid was not understood until the early part of this century when the German scientist Einstein and the Russian scientist Smoluchovski (working independently) used thermodynamic arguments to explain both the scattering from the liquid solvent and the excess scattering upon the addition of polymer as a solute. Briefly, they concluded that scattering of light by a transparent liquid is due to thermodynamic fluctuations in the concentration of the solvent, fluctuations which will clearly correlate with the compressibility of the solvent. It is clear then, that the degree to which a solvent scatters light correlates with the compressibility (i.e., it should be possible to predict the scattering power of a solvent from it's compressibility).

We can see from the last equation that if the constant K can be determined, then a plot of the Rayleigh ratio (divided by K) as a function of concentration, c, gives a straight line with zero intercept and slope of the molecular weight of the suspended particle, MW. Alternately, if one divides the Rayleigh ratio by Kc (as is commonly done), the result is a constant giving the molecular weight. It is straightforward to show that in the event of a polydisperse polymer sample, the molecular weight obtained is the weight average molecular weight. But, as is indicated at the bottom of the slide, this analysis is only sufficient if the "size" of the particle is about 20 times smaller than the wavelength of the incident light. For larger particles, Rq/Kc is NOT constant, but varies with angle from the incident direction. To account for this, Rayleigh's analysis must be extended. (In so doing, we will make explicit the meaning of "size".)

Rayleigh Scattering
large (noninteracting) molecules for Rg~l ,  for small concentrations (no interactions)

While we will not perform this derivation in any more detail than the previous one, the gist is that the amount of light scattered at an angle Q from the incident direction will be less than expected due to the slight destructive interference of light scattered by various atoms of the polymer molecule. The effect on the scattering is to multiply the previous equation for the Rayleigh ratio by the expression Si,j sin(qRij)/(qRij), where i and j label two scattering elements of the polymer molecule, Rij is the distance between them, and q is the scattering wavevector of the light. Expanding the sin function in a Taylor series and truncating higher order terms (appropriate only if qRij << l) results in a summation that can be recognized as the definition of the radius of gyration, Rg. The resulting expression for the Rayleigh ratio, Rq, (again divided by Kc) can be seen to be linear in q2 with an intercept of MW and slope of Mw Rg2/3. This form for Rq makes it easy to see that, in the event that the dissolved polymer is polydisperse, the radius of gyration obtained by a light scattering experiment will be the Z-average radius of gyration, while the molecular weight obtained is still the weight average (right?). This difference must be borne carefully in mind when trying to reconcile the two parameters.

If the Rayleigh ratio (normalized by Kc) is plotted as a function of q2, this is referred to as a Rayleigh plot. However, it was found experimentally that if the inverse of this (Kc/Rq) is plotted as a function of q2, the resulting expression, given on the second slide, remains linear over a much larger range of polymer sizes (Rg). The reason is quite subtle, but can be seen if you consider the terms in the Taylor series that were truncated. Such a plot, Kc/Rq versus q2, is referred to as a Zimm plot, after the polymer theorist Bruno Zimm who recommended it. The paper (published in 1948) in which Bruno Zimm recommended this type of data reduction did not deal as with angular dependence, however, which was understood, but with the concentration dependence. In order to get a detectable signal, concentrations of polymer must be high enough that interactions between polymers cannot be ignored. By analyzing the thermodynamics of polymer solutions, Zimm was able to show that Kc/Rq is given by the expression shown below.

Rayleigh Scattering
large interacting molecules
Zimm Plot

Zimm Equation A2 - second virial coefficient

From Laser Light Scattering of Polymer solution, obtain:

· M - mass of polymer

· Rg - Radius of Gyration (size of polymer)

· A2 - Interactions between polymers

As it stands, the Zimm equation shows that light scattering depends as given on the molecular weight, radius of gyration and second virial coefficient (A2) of the polymer in question. If the Zimm equation is considered at zero scattering angle, the inverse Rayleigh ratio can be seen to be d(p/RT)/dc = d(c/M + A2c2)/dc = 1/M + 2A2 c, which is analogous to the compressibility of a virial gas, where the osmotic pressure of the polymer solution plays the role of the gas pressure. That is, the scattering of light by a polymer in solution is due to the osmotic compressibility of the polymer solution. So, polymers that interact strongly, such as physically large molecules or highly charged polyelectrolytes, will have a large virial coefficient, and thus greatly suppressed scattering. Above you see a sample Zimm plot. What is shown is the reciprocal Rayleigh ratio as a function of sin2q + kc, where k is arbitrary (and meaningless) and chosen to adequately disperse the various angular data sets. Each plot extrapolated to zero angle gives a point on the line Kc/Rq = 1/M + 2A2 c. Similarly, each data set extrapolated to zero concentration gives a point on the line Kc/Rq = 1/M + (qRg)2 / 3. The former can be fit to give M and A2, and the latter to give M and Rg. The two values for M can be compared to ensure consistency. This data for a polymer sample is important in and of itself. But used properly, light scattering has a great many applications, some of which are described here. (If you're curious, the data shown is from a program that investigated the aggregation of sodium dodecyl sulfate (SDS) micelles onto the polymer poly(vinyl pyrollidone) (PVP). Some of the results of that program will be shown later).

Zimm Plot
Conformational Change

iota-carageenan (polysacharride)
Random Coil Conformational change to Helix ? This sets up a problem in the conformational changes of the polysaccharide i-carageenan (found in seaweed and Irish moss, and used as a thickener in ice cream, salad dressing, toothpaste, and paints). It is known that under certain conditions (of temperature and salt concentration), that i-carageenan assumes a random coil configuration (how do you suppose that was demonstrated?), and that when those conditions were varied (different temperature or different salt concentration), a conformational change to a helical structure was observed. What was not known (at least, there was no agreement) was whether the helical structure was each molecule wrapping around itself or two molecules wrapping around each other. To attempt to address this issue, Zimm plots were prepared for the polymer under each of the conditions.

Zimm Plot
Conformational Change

iota-carageenan (polysacharride)

Random Coil

Zimm Plot Results:
M = 397·103 g/mol
Rg = 994 Å
A2 = 10-2 cm3-mol-g-2

Conformational change to Helix

Zimm Plot Results:
M = 1000·103 g/mol
Rg = 2000 Å
A2 = 2.3·10-3 cm3-mol-g-2

The results suggest strongly that the in helical configuration under the conditions we explored, i-carageenan consists of two molecules coiled together. The molecular weight is approximately double, as is the radius of gyration. (And note that this is also evidence of a change in configuration, since doubling the mass of a random coil does NOT double the radius of gyration). We won't discuss it in detail, but the second virial coefficient (considered with the other data) also suggests a change from random coil to another configuration. It is of concern that the helix molecular weight is not exactly twice the random coil. Why might that be? (The question suggests, correctly, that there is a research project waiting to happen here – for example, light scattering results on the helical configuration plotted on a Holtzer plot would help clarify this whole issue). From Analysis of Mass/Time curve, discover:

·absolute scission rate

·polymer structure

·enzyme action

(random, endwise, multiple phases)
Above, I indicate qualitatively what is observed if the polymer in question is broken down as the light scattering measurement takes place. The scattered irradiance will decrease if the broken halves move apart on a rapid time scale (why does that matter?). The molecular weight can then be monitored as a function of time. A great many things can be learned about the degrading agent and polymer being degraded by properly analyzing the time dependence of the molecular weight. I'll limit this to one particular application – enzymology. Upon the addition of an enzyme to the polymer solution, the initial decrease in MW gives the initial degradation "velocity".

Time-Dependent Laser Light Scattering

Application to Enzymology

Rapid Determination of Henri-Michaelis-Menten parameters Plot of (reaction rate)
versus (substrate concentration), with fitted curve.

Plotting this initial degradation rate (normalized by the enzyme concentration) as a function of the substrate concentration allows one to obtain Henri-Michaelis-Menten parameters. Above, I show such data for the digestion of hyaluronic acid by hyaluronidase along with a fit to the Henri-Michaelis-Menten equation and the parameters from that fit.

Light Scattering and Aggregation Monitor Aggregation due to: ·Electrostatic effects ·Entropic effects ·Hydrophobic effects From Analysis of Mass/time curve, discover:

·does aggregation occur?

·on what time scale?

·in what environment?

One can also imagine that the polymer molecules, for whatever reason, aggregate. As is suggested here, one can also monitor this over time. As an example, two oppositely charged molecules were introduced into water, and the molecular weight monitored. It was observed that there was a prompt aggregation, giving molecular weights of about 10 million g/mole, followed by a slower aggregation of these into conglomerates of about an order of magnitude larger molecular weight. But, one can also simply consider the steady state of the conglomerates.
Polymer/Micelle Complex Formation

Poly(vinyl Pyrolidone) mixed with Sodium dodecyl sulfate - As SDS attaches to polymer, molecular weight of aggregate increases until saturation is reached.  low [SDS]             Intermediate/high [SDS]

In this figure, I show the scattered irradiance (divided by K) for PVP molecules dissolved in water, as a function of the concentration of added SDS. It is observed that the amount of light scattered increases approximately linearly with the SDS concentration until some concentration at which the scattering levels off and remains fixed, even though more SDS is added. The interpretation of this is that the scattering from the PVP molecules is enhanced by aggregating SDS (thus increasing the molecular weight of the molecule) until the molecules are saturated (quite literally have no more room), at which point the scattering saturates. (The lowest curve is a control showing the scattering from SDS alone in water, indicating that the changes in the other curves are not simply additional light scattered from SDS).

Polymer/Micelle Complex Formation

Poly(vinyl Pyrolidone) mixed with Sodium dodecyl sulfate - Below saturation, fraction of SDS bound to polymers approximately constant
- Above saturation, total bound SDS, f · [SDS], approximately constant.  low [SDS]                             Intermediate/high [SDS]

If you carefully analyze the data from the previous slide, you can extract the fraction of SDS added that actually winds up bound to a PVP molecule. This is shown plotted as a function of the concentration of SDS added. It is observed that the fraction of bound SDS is approximately constant (and less than 1) until the point of saturation is reached, at which point the product f · [SDS] (which is the total bound SDS) remains fixed. The fact that the bound fraction is less than one suggests that the free energy cost of binding is of the order of the thermal energy, kBT. This in turn indicates that the binding energy between micelles and polymer is of the same order as the entropy cost of binding micelles rather than letting them be free. It is not shown here, but it is also possible to obtain from this data the number of binding sites available on the PVP molecule for SDS micelles.

Polymer/Micelle Complex Formation

Poly(vinyl Pyrolidone) mixed with Sodium dodecyl sulfate - As charged SDS attaches to polymer, nominally uncharged PVP becomes charged. At low ionic strength, strong electrostatic interactions suppress scattering.  low [SDS]                                         Intermediate/high [SDS]
and/or High Ionic Strength                                 AND Low Ionic Strength

This again shows light scattered from a solution of PVP in water, as a function of added SDS concentration. The difference here is that the polymer concentration is higher and the salt concentration is varied (in the previous data, the salt concentration was kept high). For the highest salt concentration (0.5 M), the results are as before (although not enough SDS is added to see the saturation effect). But for lower levels of added salt, it is seen that after an initial increase, due to the increasing mass of the aggregate, the amount of light scattered saturates and even drops. Note also that the amount of decrease is directly dependent on the amount of added salt. This behavior reflects the fact that, while the polymer PVP is neutral, the aggregated SDS is charged. Thus, the addition of SDS to the polymer solution charges the polymer as aggregates form. These charges cause strong electrostatic interactions between the molecules, which effectively increases A2. And recall that an increase in A2 decreases the amount of light scattered. This increase of scattering as salt is added is typical for polyelectrolytes, and is due to the "screening" of the electrostatic interactions by the salt ions.

Finally, light scattering can be interfaced with a standard HPLC (sometimes translated as High Pressure Liquid Chromatography or High Performance Liquid Chromatography - no big deal, really, since it IS the high Pressure that gives the high Performance - but I digress) apparatus to greatly enhance the power of each.

Size Exclusion Chromatography with Coupled Light Scattering and Viscosimetric Detectors Obtain a complete, absolute separation and characterization of polymers. No calibration of columns necessary.

This shows a typical (but not the only) configuration for such an arrangement. With three detectors, this would be called "triple detection". Not shown is the pump that drives the flow (it would be located off the top of the page). The column fractionates the polymer sample, nominally by hydrodynamic volume, although this is not always the case. The refractive index detector gives the polymer concentration (mass per unit volume) for the fractions as they pass. Another common detector that plays this role would be a ultraviolet absorbance, or UV detector. However it is determined, the concentration is necessary for the other detectors to be of use. For example, to obtain polymer information from the viscometer, one needs the intrinsic viscosity, which depends on the concentration. Also, the light scattering apparatus directly gives only the Rayleigh ratio; the concentration is needed to obtain M and Rg. Note also that to obtain these values properly, the viscosity of a given fraction must be matched with the concentration and molecular weight of that fraction. But the detectors will take data for a given fraction at different times, as the fraction flows through the system. This difference in measurement time is called the "dead volume" and refers to the amount of fluid volume between the detectors (for a given flow rate, this volume will correspond to a delay time). All results of the triple detector apparatus are sensitive to this dead volume, some more than others. Thus, they must be carefully measured. But having done so, the triple detection technique is exceedingly powerful.

Summary

Time Resolved Laser Light Scattering

Fast, simple; / Can be coupled with SEC, viscometry techniques for:
• Polymer characterization (molecular weight, Rg, Rh, polydispersity, structure, association number)