9 Molecular Characterization Using a Unified Refractive Index-LightScattering Intensity Detector Robyn Frank, Lothar Frank, and Norman C. Ford* Downloaded by FUDAN UNIV on January 13, 2017 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/ba-1995-0247.ch009
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L a r k Enterprises, 12 Wellington Street, Webster, MA 01570
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Precision Detectors, Inc., 160 Old Farm Road, Amherst, MA 01002
We develop series expansions useful in extrapolating two-angle light-scattering data to 0° using the Debye expression for the form factor of a Gaussian coil. Errors that would be encountered if the equations were used to analyze data on molecules of other shapes are discussed. Graphs show the percent error in radius of gyration and molecular weight (M ) for hard spheres, rigid rods, and flexible rings over the range of R = 0-150 nm. Finally, we present experimental data showing that instrument calibration done in one solvent can be used in other solvents with different index of refraction and that accurate values of dn/dc can be obtained using a commercially available refractive index detector.
(R ) g
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JLJIGHT-SCATTERING INSTRUMENTS designed to be used on a routine basis
as detectors in chromatography systems are commercially available. T h e instruments measure scattered intensities at two or three angles and are used i n conjunction w i t h a concentration detector (often a differential refractometer) to determine the molecular weight ( M ) distribution and, for larger molecules, radius of gyration ( R ) . A number of questions arise i n obtaining optimum results from these detectors: (1) H o w should the calculations of M and R be done?; (2) O v e r what range of M and R are the results reliable?; and (3) Does the calibration extend to other solvents and polymers? M o u r e y and C o l l (I) suggested that the form factor for a Gaussian coil be used to analyze two-angle data at 15° and 9 0 ° to obtain M and w
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* C o r r e s p o n d i n g author
0065-2393/95/0247-0109$12.00/0 © 1995 American Chemical Society
Provder et al.; Chromatographic Characterization of Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1995.
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CHROMATOGRAPHIC CHARACTERIZATION OF POLYMERS
R . T h e y used an iterative approach to make this calculation and showed that good agreement w i t h the expected values for polystyrene standards i n tetrahydrofuran ( T H F ) were obtained for M from 1.06 Χ 1 0 to 2.3 Χ 1 0 D a . T h e y also obtained values for R i n good agreement w i t h literature values over the range of —12 to ~ 7 2 n m . First, w e provide series expansions that accurately allow the cal culation of M and R using the method of M o u r e y and C o l l without requiring an iterative approach. Second, w e present experimental data showing that a calibration using a single standard i n one solvent can be used to make measurements i n a variety of solvents. W e do this b y studying two polystyrene standards dissolved i n five different solvents w i t h specific refractive index (RI) increments (dn/dc) ranging from 0.0615 to 0.224. g
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Downloaded by FUDAN UNIV on January 13, 2017 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/ba-1995-0247.ch009
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Series Expansions for 1 / P ( 0 ) and R
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T h e usual starting point for discussion of light-scattering intensity (2) is the equation io _ 4w n V (dn/dc) 2
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sin
.AfXV
I
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êcM P(e) w
= s p(e) 0
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(1)
where i is the intensity of light i n the scattered field, I the intensity of incident light, η the solvent index of refraction, V the illuminated sample volume, ê the angle between polarization direction and scattering d i rection, c the concentration, JV Avogadro's number, λ the light wave length i n vacuum, and r the distance from the scattering volume to the detector. Ρ(θ) is the form factor depending on the scattering angle, Θ. It is equal to 1.0 for molecules m u c h smaller than λ / η and decreases w i t h increasing molecular size. S is the intensity that w o u l d be obtained i f the measurement were made at a scattering angle of 0 ° . W e have spe cialized equation 1 for linearly p o l a r i z e d light and for values of c suffi ciently low that virial coefficients may be neglected. T h e light-scattering geometry is shown i n F i g u r e 1. M o d e r n light-scattering photometers use a small diameter laser beam as a light source. I n this case, w e can replace I V w i t h P 1, where P is the laser power output and 1 the path length of the laser beam that actually contributes to the detected signal. Calculation of the detector response also requires integration over b o t h the area of the detector and the length of the laser beam contributing to the detector signal. I n practice, it is more convenient to calibrate the instrument using a single well-characterized narrow distribution standard rather than attempting calibration from a knowledge of the scattering geometry. T h e instrument used i n these studies measures light scattered at two angles. F i g u r e 2 shows the optical arrangement of the detector. U s i n g Q
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Provder et al.; Chromatographic Characterization of Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1995.
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F R A N K ET AL.
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Molecular Characterization by RI-LSD Polarization Direction Detector
θ
= Scattering Angle
φ - Angle between Polarization & Scattering Directions r
= Distance from Sample to Detector
i
= Length of Scattering Region
Downloaded by FUDAN UNIV on January 13, 2017 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/ba-1995-0247.ch009
Figure 1.
Diagram showing the light scattering geometry.
a F o u r i e r lens optical system, light scattered from 14° to 16° is collected i n a single detector. A second detector collects light scattered at 90° using a flat-ended G R I N lens 2 m m i n diameter purchased from N S G A m e r i c a , Inc., Somerset, N J . T h e maximum acceptance angle (
