Light Scattering

Light Scattering�� (slides � see 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 � �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 �� Zimm Plot (K*C/R vs. sin²(q/2):�

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