Light Scattering (lecture notes)

1. Scattering of visible light by a single molecule (case of wavelength much larger than particle). Light can be considered as a wave with oscillating electric and magnetic fields. Since electrons are charged particles, the oscillating electric field of the light causes a corresponding oscillation of the electrons of the particle. Such oscillating charges act as miniature antenna, scattering some of the incident energy in all directions. According to electromagnetic theory, the amplitude of the electric field produced by an oscillating dipole arising from the use of polarized radiation, at a distance r from the dipole, and at an angle f is given by:

E_r = {(aE_o4p²sinf)/rl²] cos2p(r/l - nt) where a is the molecular polarizability

Since intensity depends on the square of the amplitude, we can equate the intensity of the scattered radiation (i) relative to the incident intensity (I_o) as:

i/I_o = 16p⁴a² sin²f / r²l⁴ for polarized radiation.

For scattering of unpolarized radiation, this reduces to:

i/I_o = [8p⁴a² / r²l⁴](1 + cos²q) for unpolarized radiation.

Note: Intensity falls off with r² and l⁴

Intensity depends on scattering angle

2. Rayleigh Scattering – scattering from N identical particles, each much smaller than l, and dilute so interference can be neglected. Amount of Light Scattered is directly proportional to product of molar mass and concentration.

i/I_o = N[8p⁴a² / r²l⁴](1 + cos²q) for unpolarized radiation.

Substituting a in terms of dn/dC (spec. refractive index increment), C, and M

a = (2n_o /4p)(dn/dC)(C/N) and noting that

N = [C(g/mL)/M]N^o.

Yields: i/I_o = [2p² n_o²(dn/dC)² / r²l⁴ N^o] CM(1 + cos²q).

Define “Raleigh Ratio” as R_q = (i_q / I_q)(r² / (1 + cos²q)) and thus

R_q = [2p² n_o²(dn/dC)² / l⁴ N^o] CM or R_q = KCM.

Note: This is frequently described in terms of the excess Rayleigh ratio, R_q , expressed as light scattered per unit of solid angle in excess of that scattered by the pure solvent; C as conc. in g/mL, M_w is the weight-average molar mass, A₂ is the 2^nd virial coefficient, and the constant K* = [4p² n_o²(dn/dC)² / l⁴ N^o], n_o is the refractive index of the solvent and dn/dC the specific refractive index increment. P_q is a form factor that describes the angular dependence from which the mean square radius <r_g²> may be determined.

(K*C / R_q) = 1/M ideal solutions or (K*C / R_q) = 1/(M P_q) + 2A₂C …for real solutions

Note: Different experimental methods yield different types of experimentally arrived at “Molecular Weights.” Light scattering yields a “weight average” molar mass– solutions must be scrupulously clean since dust will contribute to average as very large molecules.

SC_iM_i^y / SC_iM_i^y-1

Number Average M_n SN_iM_i / SN_i SC_i / S(C_i /M_i) y = -1 Osmotic Press / F.Pt.

Weight Average M_w SN_iM_i² / SN_iM_i SC_iM_i / SC_i y = 0 Light Scatt. / Sed. Eq.

“Z” Average M_Z SN_iM_i³ / SN_iM_i² SC_iM_i² / SC_iM_i y = 1 Sedimentation Equil.

3. Dynamic Light Scattering – Measurement of Diffusion to get R_h

The movement of molecules in solution is related to their diffusion constants or frictional coefficients

D = kT/f = (RT)/( N_of)

Consider the scattering from a small volume element of solution at some local concentration. As molecules move into and out of this volume element, the local scattering will vary with time. Dynamic Light Scattering measures these fluctuations with time.

Di_t = i_t – av.<i>

Do an autocorrelation analysis of variation of Di_t at time t vs. (t + t)

A(t) = average product of (Di_t · Di_{(t + t)})

(Note: A(t) will be large when t = 0 and A(t) = some minimum at large t, with the

fall off depending on “D” – the diffusion coefficient)

It can be shown that ln(A(t)) = ln(A_o - ([8p²n²/l²][sin²(q/2)])·D·t

Thus a plot of ln(A(t)) vs. t can yield the diffusion coefficient “D”

Normally report “Hydrodynamic Radius – R_h” ; D_o = kT/f = kT/(6phR_h)

4. X-ray Scattering – With shorter wavelengths, scattering occurs from particles that are comparable or smaller than the incident radiation. Radiation will induce dipoles within the large particle that are out of phase with other parts of the large molecule causing appreciable interference effects. This type of scattering information can yield information about size and shape as well as mass of the particle in terms of the R_G (Radius of Gyration) of the particle.

Define P(q) = (scattering by real particle at q)/ (scattering by point particle at q)

At low angle; P(q) ~ (1/n²) SS 1 - ([16p² / l²][sin²(q/2)]/6n²) SS r_ij²

or P(q) = 1 - ([16p²][sin²(q/2)]R_G² /3l²) + ; where R_G² = (1/2n²) SS r_ij²

For “real” solutions using wavelengths with large particles – and since 1/(1-x) ~ 1+x, we have

KC / R_q ~ [1/M + 2A₂C + ]·[1 + ([16p²R_G² / 3l²][sin²(q/2)])]

Analyze the light scattering data as a function of concentration and angle, and use a “Zimm Plot” to get 1/M_w by extrapolating to both low concentration and low angle.

5. Other Types of Scattering: Low-Angle X-ray Scattering / Raman Scattering –

Some Applications – Wyatt Technologies Web Site (http://www.wyatt.com/theory/index.cfm)

Headline Summary for Light Scattering

Static Light Scattering (Classical or Rayleigh Scattering): Absolute Molecular Weight

1) Amount of Light Scattered is directly proportional to product of molar

mass and concentration.

(K*C / R_q) = 1/M ideal solutions or

(K*C / R_q) = 1/(M P_q) + 2A₂C …for real solutions

2) For large or rod shaped particles or using smaller wavelengths, variation of

scattered light with scattering angle is proportional to average size (R_G²)

of the scattering molecules. For classical light scattering

Size Range: R_G > 10nm to 150nm

where R_G² = (1/2n²) SS r_ij² for n identical particles

Dynamic Light Scattering (DLS or Photon Correlation Spectroscopy): Stokes Radius (R_h)

1) Measure variation of light scattering over short times (microseconds)

– look at autocorrelation function using Stokes-Einstein Equation à “D”

- report “Hydrodynamic Radius – R_h” instead of D_o = kT/f = kT/(6phR_h)

Size range: R_h from 1 to 1000nm

- particularly good at sensing very small amounts of aggregates (<0.01% by wt.)

Practical Considerations:

1) Normally use SEC plus a 2-detector or 3-detector experiment

SEC/LS/RI or SEC/LS/RI/UV

Light Scattering - LS = K_LSCM(dn/dC)²

Refractive Index – RI = K_RIC(dn/dC)

(dn/dC) ~ 0.186 mL/g for proteins; different if carbohydrate present

Determine M from ratio of the two detectors:

LS/RI = M [(K_LS/K_RI)(dn/dC) = M K’

or M = K’(LS)/(RI); (2-detector method valid when (dn/dC) is known)

(use protein standards like BSA to calibrate both RI and LS detectors)

(use 3^rd UV detector for concentration when carbohydrate is present)

2) Polydispersity [(M_w/M_n) and (M_z/M_n)]

MW Averages - SC_iM_i^y / SC_iM_i^y-1 (or M_w = SC_iM_i / SC_i for y = 1)

Note: Since LS ~ M^.C and RI ~ C, when normalized and

checking for oligomerization – LS = RI for monomer

LS = 2 RI for dimer

LS = 3 RI for trimer, etc.

MALS – Multi-angle Light Scattering

- with large particles (or shorter wavelengths) – can obtain information on size

- Angle and Conc. dependence à Zimm Plot (K*C/R vs. sin²(q/2):

à y-intercept gives 1/M, slope gives R_G²