Chapter 8: Light Scattering

Radiation/Light Scattering Techniques:

There are many types of radiation scattering experiments which are used to characterize materials, including small angle light scattering (SALS), small angle x-ray scattering (SAXS), x-ray diffraction (XRD) (more on this when we talk about T_g and semi-crystalline polymers), and small angle neutron scattering (SANS), all of which operate off of the same basic principles but with different types of scattered radiation. Light scatting is based on the principle that an incident beam of radiation will scatter off of a sample in some predictable way as a function of the angle of the scattering detector with respect to the sample, the wavelength of the incident radiation, and the refractive index of the sample.

Principles of Radiation Scattering

When a beam of incident light (or other form of electromagnetic radiation) interacts with a molecule, in general the incident radiation can either be absorbed or scattered. The latter behavior occurs when the oscillating electric field of the radiation induces an oscillating dipole in the particle, which then acts as a source of additional radiation. Therefore, the particle then emits light in all directions, scattering the original incident light beam in all directions (on average). We will be primarily concerned with Rayleigh scattering, where it is assumed that the energy of all emitted light is the same as the incident light and thus all emitted light has the same wavelength as the incident light. The phenomenon of light scattering is responsible for most visible light; for example, when we attribute a color to the sky it is because incident light from the sun infringes upon gas particles, which then scatter the light in all directions, including toward observers on the Earth’s surface. We will see why the sky appears blue, rather than other colors, in a minute.

Light scattering depends on the geometry, optical properties, and thermodynamics of the system under consideration (in this lecture we mostly discuss polymer-solvent solutions). The geometry is based on the distance of the detector from the sample (r) and the angle of the detector with respect to the incident light (θ). The further away the detector is, the lower the intensity of the incident light. The optical properties of the system are based on the wavelength of the incident light (λ), index of refraction n (or polarizability α) of the polymer solution and sample, and the derivative of the index of refraction with respect to solution concentration . The important thermodynamic properties are the concentration c₂ of the polymer solution, the weight-averaged molecular weight M_w, and the second virial coefficient, which is again related to χ from Flory-Huggins.

The basics of light scattering come from the scattering equation, which we will derive in a bit, but physically we should note that scattering arises from differences in variations in solution properties that differ based on the incident radiation. For light scattering, we measure the difference in polarizability ∆α; for X-ray scattering, the difference in electron density ∆ρ; and for neutron scattering, the difference in neutron scattering length ∆b. The choice between which type of scattering to employ depends on what resolution you want, and the ease of determining these different properties for different polymer systems. More precisely, we measure fluctuations in the polarizability of the entire solution (polymer + solvent) that emerge from the difference in polarizability between the solution by itself and the polymer by itself. In solution, the difference in properties between these two components give rise to different scattering behavior from different regions of solution corresponding to the relative concentration of polymer throughout the solution; as polymer diffuses throughout the solution, we can thus measure fluctuations of the polarizability to gain information about the polymer itself, knowing the properties of the background solvent.

In essence what we want to achieve with light scattering is to shine a light on the polymer solution and observe the intensity of the scattered light. We then want to see if there is a difference in polarizability between the polymer and the solvent. If there is a difference then we can relate the observed scattering intensity to concentration fluctuations of the polymer chain which can then be related to the second virial coefficient due to osmotic pressure build up from those fluctuations.

Detailed Light Scattering Derivation:

Theory of Light Scattering from Point Particles:

To derive the scattering equation, let’s first consider a simpler system: a dilute gas in vacuum, filled with small particles. We will call these small particles point scatterers, implying that they have no angular dependence on scattering since they have no effective size. To be more precise, the assumption of no size just says that the size of the particle is much smaller than the wavelength of the radiation. We start by considering only a single particle.

As discussed before, when a light wave encounters a particle, the oscillating electric field of the radiation forces the electron and nucleus of the particle to move in opposite directions (due to their opposite charges), thereby inducing a dipole that oscillates with the incident wave. The magnitude of this induced dipole is related (by definition) to the polarizability, α, of the particle. Since oscillating dipoles emit radiation, the particle thus radiates light away from the particle in all directions with an intensity given by

Here, θ is the angle between the incident radiation and the detector, λ is the wavelength of the incident radiation, α is the polarizability of the point particle, and I₀ is the initial intensity of the incident radiation. The fourth order dependence on shows that the smallest wavelengths will actually be scattered most (this same

relation determines why the sky is blue, since blue is the smallest visible wavelength and hence scatters very strongly from gas particles in the atmosphere, and scattered light is what we primarily observe visibly). By scatters most we mean that the intensity at any given θ will be highest when the wavelength λ is lowest. To generalize this equation to many particles, it is only necessary to scale by the density of point scatterers in solution

Here, N is the number of particles (not to be confused with the degree of polymerization of a polymer), and V is the total volume of the solution. The assumption is effectively that the scattering between different point particles is uncorrelated, and we can simply sum the contributions from all particles in a given volume to get the total intensity (per unit volume). Next, we can relate the polarizability (which is not easily determined) to the dielectric constant of the medium using

Again, we give the effective polarizability of the medium as effectively the concentration of particles times α. We also know that the index of refraction is given by

and for a gas is given by the following relationship

which is essentially an expansion of the index around its known value of 1 in a gas. Note that for polymers, the expression is the same, except that instead of expanding around 1 the value is instead replaced with the solvent index of refraction n₀. In other words, we again make the assumption that we can relate the index of refraction of a solution to the index of refraction of the majority component (vacuum in the gas case, solvent in the polymer-solution case) to the change in index of refraction with concentration multiplied by the concentration. Squaring the index of refraction thus gives an expression for the polarizability

This gives us the polarizability of the sample in terms of quantities that we can measure more easily, specifically the concentration of the (gas/solution) and the change in index of refraction with respect to concentration. It is possible to measure the change in index of refraction with concentration dn/dc directly by increasing the concentration of a solution and measuring the change in the angle of refraction for that solution as a result of the change in concentration, since recall that the index of refraction determines the angle of refraction from Snell’s law

So the relation above provides for a simple way of actually measuring polarizability. The total intensity of the scattered light can now be related to only scattering geometry and measurable optical parameters, and is given by combining equations (1),

(2) and (6) to yield the complicated expression

Canceling out terms and simplifying

To further simplify the expression, we first relate N/V to concentration via the relation $\frac{N}{V} = \frac{c}{M/N_A}$ and we get

This equation alone is called the Rayleigh Equation for ideal elastic scattering, and gives the intensity of scattered light as a function of the geometry of the sample (θ, r), the characteristics of the solution (dn/dc, M, c) and characteristics of the incident light (λ, I₀). It is also convenient to define the Rayleigh ratio:

The advantage of looking at the Rayleigh ratio is that the constants related to scattering geometry are incorporated in the left hand side and R is therefore effectively independent of scattering geometry. We can reduce this expression to

where K is the optical constant consisting of the combination of a large number of constants that are properties of a given choice of incident ratio and a given sample choice. Writing out K (for a polymer, with the corresponding inclusion of background index of refraction n₀ as measured above) gives:

All of the parameters of the optical constant are either determined from the experimental set up (e.g. λ) or can be measured from the properties of the sample (dn/dc). We also know the concentration c of the polymer sample. Finally, note that the Rayleigh ratio itself can be measured from the scattering intensity measured from a sample. Hence we have a simple (or at least, simplified version) equation relating the relationship between the Rayleigh ratio and the mass of the point scatterers (or polymers for n₀ > 1) M!

Light Scattering from Polymer Solutions:

The derivation above assumed a dilute gas (though we made minor corrections to account for a non-vacuum background when using the background index of refraction n₀). In a liquid sample, the concentration of molecules will be much greater than in a dilute gas, and as result there will be the potential for interference between scattered light that eliminates any measurable scattering. Thus, in a uniformly dense liquid, we may not expect scattering because of a uniform polarizability - however, in reality, liquids are not uniformly dense! Instead, because of mass fluctuations in an otherwise uniform distribution of liquid particles, at any given instant in time different regions of a liquid will have a different number of molecules, leading to a net scattering if we assume that mass fluctuations are uncorrelated (that is, at any given time any region can have a slightly higher or lower mass independent of other regions). The key idea to remember is that the scattering from any given isolated region of the liquid will depend on that region’s density (again referring to equation (2)), and in any system

with free motion (such as a liquid or gas) there will be local density fluctuations that give rise to net scattering. We can thus measure the Rayleigh ratio from even a uniform liquid, which will be a key idea in analyzing polymer solutions.

In the case of a dilute solution of polymer and solvent, we will find that we can relate the Rayleigh ratio of the polymer alone to properties of that polymer, and we thus want to measure just the scattering of the polymer. However, in practice we will only be able to measure the scattering (and thus the Rayleigh ratio) of the entire solution, so we define the difference in Rayleigh ratios as:

We will now discuss how to calculate this difference in Rayleigh ratios between the solution and solvent alone, yielding what is effectively the change in scattering due to just the polymer molecules. We will find that this light scattering depends on the difference in polarizability between the polymer coils and the background solvent; if there were no difference in polarizability we would be unable to distinguish polymer from solvent. We will now discuss dilute polymer solutions in more detail.

Let’s consider a dilute polymer solution. There are four features of this system that need to be considered that were not applicable in the ideal gas system discussed in the previous section.

1. The background solvent will experience density fluctuations and will tend to scatter light on its own. We have to subtract this as background scattering
2. The polymer solution will not be uniformly distributed and will experience concentration fluctuations leading to strong scattering which is amplified by the difference in polarizability between polymer and background solvent
3. The build up of polymer concentration in a given area is opposed by resulting osmotic pressure. Osmotic pressure will drive uniform polymer distribution and oppose fluctuations.
4. We cannot treat polymers as simple point scatters as the size of a polymer coil will be on the order of the incoming wavelength. This will lead to interference between the light scattered within the same coil.

Let’s go ahead and deal with each of these issues. First, we have to get rid of the background effects of the solvent. We already effectively take this into account by considering ∆R as noted above. Again, R is the Rayleigh ratio as defined in equations (11) and (12). Next, we note that any remaining scattering, aside from the pure solvent, comes from fluctuations in polarizability throughout the sample, which we will later relate to concentration fluctuations of polymer. If we imagine dividing the entire system in N regions of volume δV , then N/V = δV , we can relate

the original expression we derived for the intensity of scattering to the Rayleigh ratio using:

Having related the Rayleigh ratio to polarizability, we now can express the difference in the Rayleigh ratios to the fluctuations in polarizability, recognizing that fluctuations are time-averaged

Next, we recognize from equation (6) that there is a relation between polarizability and concentration of polymer, so we can also relate their fluctuations

Finally, we substitute in this expression for polarizability fluctuations into the expression for the difference in Rayleigh ratios to obtain an expression that instead refers to concentration fluctuations:

Recall that concentration fluctuations refer to the fluctuations in concentration of polymer around some equilibrium value (i.e. the overall concentration of polymer in a well-mixed solution), which is characteristic of any system even at equilibrium; however, any fluctuation that drives a system away from its equilibrium value will give rise to a thermodynamic force that drives the system back to equilibrium. In the case of concentration fluctuations, the thermodynamic driving force that drives a system toward an equilibrium concentration is the osmotic pressure. Thus, we can relate the magnitude of concentration fluctuations to the change in the osmotic pressure as the concentration gets farther away from its equilibrium value; an expression for the concentration fluctuations can thus be derived that is called the Einstein-Smoluchowski relation

Note that in this equation R is the ideal gas constant, NOT the Rayleigh ratio. The equation effectively relates the average magnitude of fluctuations (written as h(δc₂)²i) to the concentration of the solution divided by the change in osmotic pressure with the change in concentration. This derivative is expected since it reflects the rise in osmotic pressure as we are driven away from equilibrium by concentration fluctuations.

This general relation is true for any solution; however, since we know we are dealing with polymer solutions, we can further simplify the Einstein-Smoluchowski relation based on the expression for osmotic pressure we developed from Flory-Huggins theory. There, we wrote the osmotic pressure as:

where $A_2 = \left ( \frac{1}{2} -\chi \right )\frac{V_1x_2^2}{M^2}$ is the second virial coefficient, and taking the derivative gives

Again, in these equations R is the ideal gas constant (it’s unfortunate that this notation is confusing by convention). Combining equations (20), (21) and (24) together and simplifying with the same optical constant K defined in equation (13) gives:

Here, I have omitted any terms in the expression for osmotic pressure beyond the first order term (with A₂ as the prefactor). If we look at this equation, we see that if the second virial coefficient A₂ is equal to 0, then we get the expression ∆R = KMc₂ exactly the expression for the Rayleigh ratio we derived for an ideal gas in a vacuum! This observation is consistent with the knowledge that A₂ = 0 occurs when χ = 1/2 and we have a theta solution, where the polymer is ideal. Since A₂ is an effective measure of the strength of interactions between polymers in solution, we see that as interactions become stronger (the magnitude of A₂ increases) ∆R becomes smaller; this is consistent with strong interactions reducing fluctuations and thus contributing more order to the solution, masking scattering by destructive interference.

Finally, we also have to note that our assumption of point scatterers is not accurate for large polymer coils, since in general the size of a coil is on the order of 100-1000 ˚Aand the wavelength of incident light(e.g. lasers) is on the order of 5000 ˚Aas well. When the polymer molecule is at this size, it is possible for light to scatter off of different parts of the same molecule, which could give rise to interference effects (effectively it’s like having a uniform concentration of scatterers inside the volume occupied by a single coil).

To take this effect into account, we define an angle-dependent factor P(θ). This function is basically related to the connectivity of polymer chains, and more specifically related to the fact that fluctuations in polymers are correlated due to connectivity (rather than uncorrelated as assumed earlier in the uniform scatterer case). We then rewrite our definition of the excess Rayleigh ratio as:

Debye showed that for Gaussian polymer coils, P(θ) is given by:

This allows us to bring in the radius of gyration as a measurable quantity! In the limit of very small θ, we can simplify this equation to:

Putting this together with the equation for the difference in Rayleigh ratio yields (after slight rearrangement):

While this equation is quite complicated, it shows that we can relate the optical constant, excess Rayleigh ratio and concentration of the polymer solution, all of which are experimentally obtainable, to the second virial coefficient (related to χ), radius of gyration of the polymer coils, and molecular weight of the sample (in reality a molecular weight average but we neglect that here), all of which are important properties we have used repeatedly.

Zimm Plots: Extracting Useful Data From Light Scattering

The equation for the excess Rayleigh ratio is complicated, but it is possible to extract useful data from it using a Zimm plot. The basic idea behind a Zimm plot is to note that $\frac{Kc_2}{\Delta R}$ is a function of effectively two variables: the first term is a linear function of c₂ while the second term is a linear function of sin²(θ/2). If we separate out each of these variables by holding the other constant, we can thus derive the 3 useful parameters outlined above ($\(\chi$, $\langle R_g^2\rangle\), and M). The procedure is to perform a double extrapolation: hold one variable (either c₂ or θ) constant while calculating the other variable and extrapolating it to 0, then doing the same for the other variable.

More explicitly, we can see that extrapolating θ to 0 gives an equation for c₂ of:

Figure 11. Zimm Plots Experimental Method.

Graphing with respect to c₂ thus gives the second virial coefficient from the slope and $\frac{1}{M}$ from the intercept. Conversely, extrapolating c₂ to 0 gives:

Thus, graphing this expression with respect to sin²(θ/2) yields a line with a slope related to $\frac{\langle R_g^2\rangle}{M}$ and an intercept is related to $\frac{1}{M}$.