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:
����������������������� Er = {(aEo4p2sinf)/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 (Io) as:
����������������������� i/Io� = 16p4a2 sin2f / r2l4���� for polarized radiation.
For scattering of unpolarized radiation, this reduces to:
����������������������� �� i/Io� = [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/Io� = N[8p4a2 / r2l4](1 + cos2q)��� for unpolarized radiation.
����������� Substituting a in terms of dn/dC (spec. refractive index increment), C, and M
a� = (2no /4p)(dn/dC)(C/N) and noting that
������������� N =� [C(g/mL)/M]No.
Yields:� i/Io� = [2p2 no2(dn/dC)2 / r2l4 No] CM(1 + cos2q).
����������� Define �Raleigh Ratio� as���� Rq� =� (iq / Iq)(r2 / (1 + cos2q)) and thus
Rq�
=� [2p2 no2(dn/dC)2 / l4 No] CM� or� Rq�
=�� KCM.
Note: This is frequently described in terms of the excess Rayleigh ratio, Rq , expressed as light scattered per unit of solid angle in excess of that scattered by the pure solvent; C as conc. in g/mL, Mw is the weight-average molar mass, A2 is the 2nd virial coefficient, and the constant K* = [4p2 no2(dn/dC)2 / l4 No], no is the refractive index of the solvent and dn/dC the specific refractive index increment.� Pq is a form factor that describes the angular dependence from which the mean square radius <rg2> may be determined.
���������
���� (K*C / Rq) =� 1/M�� �ideal solutions or�� (K*C / Rq)� =� 1/(M Pq) �+ 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.
����������������������������������������������������������������������� �� SCiMiy / SCiMiy-1
Number Average� Mn������� SNiMi / SNi����������� SCi / S(Ci /Mi) ��� y = -1��� Osmotic Press / F.Pt.
Weight Average� Mw�������� SNiMi2 / SNiMi ���� SCiMi / SCi �������� y = 0����� Light Scatt. / Sed. Eq.
�Z� Average� MZ�������������� SNiMi3 / SNiMi2���� SCiMi2 / SCiMi �� y = 1����� Sedimentation Equil.
3. Dynamic Light Scattering � Measurement of Diffusion to
get Rh
The movement of molecules in solution is related to their diffusion constants or frictional coefficients
����������������������������������������������������������� D = kT/f� = (RT)/( Nof)
����������� 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.�
����������������������������������������������� Dit� =� it � av.<i>��
����������� ��� Do an autocorrelation analysis of variation of Dit at time t vs. (t + t)
����������������������� A(t)� =� average product of (Dit � 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(Ao� -� ([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 � Rh� ; Do = kT/f = kT/(6phRh)
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 RG (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 rij2�����������
�������� or�� P(q) =� 1 - ([16p2][sin2(q/2)]RG2 /3l2)� +��� ; where� RG2� =� (1/2n2) SS rij2��
� For �real� solutions using wavelengths with large particles � and since 1/(1-x) ~ 1+x, we have
����������� KC / Rq� ~� [1/M �+ 2A2C� +� ]�[1 +� ([16p2RG2 / 3l2][sin2(q/2)])]�
����������� Analyze the light scattering data as a function of concentration and angle, and use a �Zimm Plot� to get 1/Mw 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 / Rq) =� 1/M�� ��������������������ideal solutions or�
�(K*C / Rq)� =� 1/(M Pq) �+ 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 (RG2)
of the scattering molecules.� For classical light scattering
�
������������ ���������������������� where RG2� =� (1/2n2) SS rij2�� for n identical particles
Dynamic Light Scattering (DLS or Photon Correlation Spectroscopy): Stokes Radius (Rh)
����������� 1) Measure variation of light scattering over short times (microseconds)
� look at autocorrelation function using Stokes-Einstein Equation � �D�
- report �Hydrodynamic Radius � Rh� instead of� Do = kT/f = kT/(6phRh)
����������������������������������� Size
range:�� Rh 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 = KLSCM(dn/dC)2
����������������������� ����� Refractive Index � RI = KRIC(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 [(KLS/KRI)(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 3rd UV detector for concentration when carbohydrate is present)
�����������
����������� 2) Polydispersity [(Mw/Mn) and (Mz/Mn)]
MW Averages - SCiMiy / SCiMiy-1�� (or Mw = SCiMi / SCi �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 RG2�