CHAPTER
3.2 Light Scattering
3.2.3 Scattering from Large Particles
The equations to this point assume that each solute molecule is small enough compared to the wavelength of incident light to act as a point source of secondary radiation, so that the intensity of scattered light is symmetrically distributed as
A
P1 θ2
θ1
P2 C
B
(a)
Incident (b)
Scattering envelope for larger scatterers
Small scatterers direction
FIGURE 3.4
(a) Interference of light scattered from different regions of a scatterer with dimensions comparable to the wavelength of the scattered light. (b) Scattering envelopes for small and large scatterers. These scattering envelopes are cylindrically symmetrical about the direction of the incident light.
shown inFig. 3.4b. If any linear dimension of the scatterer is as great as aboutλ/20, however, then the secondary radiations from dipoles in various regions of the scatterer may vary in phase at a given viewing point. The resulting interference will depend on the size and shape of the scatterer and on the observation angle.
The general effect can be illustrated with reference to Fig. 3.4a, in which a scat- tering particle with dimensions near λ is shown. Two scattering points, P1 and P2, are shown. At planeA, all the incident light is in phase. PlaneBis drawn per- pendicular to the light which is scattered at angleθ2from the incident beam. The distanceAP1B,AP2Bso that light that was in phase atAand was then scattered at the two dipolesP1andP2will be out of phase atB.
Any phase difference at Bwill persist along the same viewing angle until the scattered ray reaches the observer. The phase difference causes an interference and reduction of intensity at the observation point. A beam is also shown scattered at a smaller angleθ1, with a corresponding normal planeC. The length differenceOP1C
2 OP2Cis less than OP2B2 OP1B. (At the smaller angleθ1, AP2.AP1while P2C,P1C, so the differences in two legs of the paths between planesAandCtend to compensate each other to some extent. At the larger angleθ2, however,AP2.AP1 and P2B.P1B.) The interference effect will therefore be greater the larger the observation angle, and the radiation envelope will not be symmetrical. The scatter- ing envelopes for large and small scatterers are compared schematically in Fig. 3.4b. Both envelopes are cylindrically symmetrical about the incident ray, but that for the large scatterers is no longer symmetrical about a plane through the scat- terer and normal to the incident direction. This effect is calleddisymmetry.
Interference effects diminish as the viewing angle approaches zero degrees to the incident light. Laser light-scattering photometers are now commercially avail- able in which scattering can be measured accurately at angles at least as low as 3. The optics of older commercial instruments which are in wide use are restricted to angles greater than about 30 to the incident beam. Zero angle intensities are esti- mated by extrapolation. It is always necessary to extrapolate the data to zero con- centration, for reasons which are evident from Eq. (3-44). Conventional treatment of light-scattering data will also involve an extrapolation to zero viewing angle.
The double extrapolation to zero θand zerocis effectively done on the same plot by the Zimm method. The rationale for this method follows from calculations for random coil polymers which show that the ratio of the observed scattering intensity at an angle θ to the intensity that would be observed if there were no destructive inference is a function of the parameter sin2(θ/2). Zimm plots consist of graphs in which Kc/Rθ is plotted against sin2(θ/2)1bc, where b is an arbitrary scale factor chosen to give an open set of data points. (It is often convenient to takeb5100.) In practice, intensities of scattered light are measured at a series of concentrations, with several viewing angles at each concentration. The Kc/Rθ (or Hc/τ) values are plotted as shown inFig. 3.5. Extrapolated points at zero angle, for example, are the intersections of the lines through theKc/Rθ values for a fixed c and variousθvalues with the ordinates at the correspondingbc values. Similarly, the zerocline traverses the intersections of fixedθ, variablecexperimental points 110 CHAPTER 3 Practical Aspects of Molecular Weight Measurements
with the corresponding sin (θ/2) ordinates. The zero angle and zero concentration lines intercept at the ordinate and the intercept equalsM2w1:
In many instances, Zimm plots will curve sharply downward at lower values of sin(θ/2). This is usually caused by the presence of either (or both) very large polymer entities or large foreign particles like dust. The large polymers may be aggregates of smaller molecules or very large single molecules. If large molecules or aggregates are fairly numerous, the plot may become banana shaped. Double extrapolation of the “zero” lines is facilitated in this case by using a negative value ofbto spread the network of points.
It is obviously necessary to clarify the solvent and solutions carefully in order to avoid spurious scattering from dust particles. This is normally done by filtra- tion through cellulose membranes with 0.2- to 0.5-μm-diameter pores.
If a laser light-scattering photometer is used, the scattered light can be observed at angles only a few degrees off the incident beam path. In that case extrapolation to zero angle is not needed and the Zimm plot can be dispensed with. The turbidities at several concentrations are then plotted according to Eq. (3-53). A single concentration observation is all that is needed if the concen- tration is low (the A3c2 term in Eq. 3-53 becomes negligible) and if the second virial coefficient A2 is known. However, A2 is weakly dependent on molecular weight and better accuracy is generally realized if the scattered light intensities are measured at several concentrations.
C = 0 LINE C1
bc5 bc6 C2 C3
C4 C5 C6 θ4
θ3
θ2
θ1 θ = 0 LINE
sin2 θ/2 + bc sin2θ4
2 Kc/RθorHc/T
FIGURE 3.5
Zimm plot for simultaneous extrapolation of light-scattering data to zero angle (θ) and zero concentration (c). The symbols are defined in the text. x, experimental points;
Kextrapolated points; x, double extrapolation.