Revealing the dn/dc of Polymer Solutions
K-D. Bures,
The refractive Index Increment (dn/dc) is of fundamental importance for molecular weight determination of Polymers using Static Light Scattering, both with classical goniometer systems for batch applications (SLS) or modern multi-angle detectors in a Chromatography environment (MALS). An incorrect dn/dc will dramatically affect the calculated molar mass obtained from Light Scattering experiments.
Static Light Scattering - also often known as MALS or LALS - is one of the most widespread methods for Molecular Weight determination both as stand-alone systems like classical goniometers as well as in combination with separation techniques like GPC (Gel Permeation Chromatography) or FFF (Field Flow Fractionation). Fundamental principal of this method is the quantification of light scattered by a sample under investigation. In a stand-alone system like a goniometer, the determined molar mass is an absolute value while systems combined with separation techniques deliver a "nearly absolute" value, since determination of instrument constants with a known standard (an isotopic scatterer, typically a Polystyrene Standard) is required for normalization prior to the measure of unknown samples. Both methods however, require knowledge of the specific refractive index increment (dn/dn) of the sample/solvent combination used in order to deliver molecular weight values and related parameters.
A glance at the Zimm equation used in Light Scattering (Eq. 1), shows how important dn/dc is, since it appears as a squared term of the Debye constant, therefore the dn/dc error will reflect doubled into the calculated molacular weight.
Eq.1: 
where Eq.2: 
A number of different literature sources are available for dn/dc values; its use however, implicates a high degree of uncertainty and is not indicated, especially in those cases where Light Scattering experiments are performed at a different wavelenght than published dn/dc values. Also, detailed comparison of published data shows a wide range of values from different sources for the same compound, which makes a selection of the "correct" dn/dc value nearly impossible, as shown in table 1 for Polystyrene in THF.
For all the above mentioned reasons, the individual determiantion of the specific refractive index of the compound under investigation is not only indicated, but imperative requirement for the correct determination of Molecular Weight with Light Scattering Methods.
In the past, interferometric type refractometers have been widely used for dn/dc determination and believed to be more accurate and reliable than common differential refractometers. New developements in the field of differential refractometers like the WGE DnDc-2010 however, have replaced and outclassed the interferometric method mainly because of their superior reproducibility of results and easier handling which leads to faster and more accurate measurements. Most differential refractometers available on the market, are optimized for GPC or HPLC applications and are not necessarily suitable for dn/dc determinations as when used as pure concentration detectors, wavelenght of the light source is not a parameter of interest for the user, and very often white light or near infrared sources are used for these detectors. However, as show by Cauchy's dispersion relation (Eq. 3), dn/dc is wavelenght-dependent, and must be measured at the same (or closest possible) wavelenght of the laser used for light scattering. For this reason it is required to apply instruments specifically designed for dn/dc determination like the DnDc-2010 instead of common Refractive Index Detectors.
Eq. 3: 
Accurate measurement of dn/dc, require first determination of the instrument constant K (see Eq. 4) which describes the optical properties of the instruments as well as electronic conversion parameters. K is usually determined by measurement of five or more different concentrations in the range from 0,1 mg/ml to 15 mg/ml of a substance with known dn/dc like KCI in Water and linear fit of calculated Dn vs. concentration. The obtained slope of the straight line will then be the instrument constant, which can be used for all following measurement independently of the solvent then used. This determination must be performed with great care, as it will affect all subsequent measurement, and it is advisable to repeat it in regular intervals in order to control the reproducibility and state of the instrument.
Fig. 1 Calibration 
Eq. 4: 
where C = Concentration and K = Instrument Constant.
Measurement of an unknown sample is then performed similarly, measuring five or more concentrations, covering at least an order of magnitude and plot of signals versus concentration, obtaining then, after calculation, the dn/dc value. A comfortable Software package for data acquisition and analysis makes manual calculations obsolete and provides a perfect platform for routine work.
Fig. 2 Measurement 
It is important to note, that a complete measurement or determination of constant, usually requires not more than 10 minutes to complete. Therefore repeat measurments can be easily achived thus allowing statistical analysis and increasing by this reliability of obtained data.
The DnDc-2010 differential refractometer in combination with the BI-DNDCW Software by Brookhaven Instruments Corp. is a powerful and reliable tool for dn/dc measurements in all common solvents at temperature up to 80 °C. Its proven precision, accuracy and reproducibility allow molecular weights determination with light scattering methods of unsurpassable precision and reliability, making usage of literature data or assumed values completely unnecessary.
Table 1: Comparison of dn/dc Literature Data for Polystyrene in THF at 546 nm
| Waveleght [µm] | dn/dc | Source |
| 0.546 | 0.193 | Jordan, E. F.; Jr. J. Polymer Sci. A-1, 6, 2209-2219 (1968) |
| 0.546 | 0.198 | Angelo, R. J.; Ikeda, R. M.; Wallach, M. L. Polymer, 6, 141-156 (1965) |
| 0.546 | 0.193 | |
| 0.546 | 0.189 | Schulz, G. V.; Baumann, H. Makromol. Chem. 114, 122-138 (1968) |
| 0.546 | 0.194 | Schulz, G. V.; Baumann, H. Makromol. Chem. 114, 122-138 (1968) |
| 0.546 | 0.195 | Schulz, G. V.; Baumann, H. Makromol. Chem. 114, 122-138 (1968) |
| 0.546 | 0.196 | Schulz, G. V.; Baumann, H. Makromol. Chem. 114, 122-138 (1968) |
| 0.546 | 0.198 | Schulz, G. V.; Baumann, H. Makromol. Chem. 114, 122-138 (1968) |
Eq. 1: Zimm Light Scattering Equation
Eq. 2: Debye Constant
Eq. 3: Cauchy dispersion relation
Eq. 3: Signal development of Refractive Index Detectors
Literature:
(1) Huglin, M., ed. Light Scattering from Polymer Solutions, Academic Press (1972)
(2) Theisen, A.; Johann, C.; Deacon, M. P.; Harding, S. E., Refractive Increment Data-Book, Nottingham University Press (2000)
(3) St. Weidner, U. Just, Certification of Reference Material BAM=P002 Polystyrene (PS), Bundesanstalt für Materialforschung und prüfung (BAM), Berlin