GPC Theory: Light scattering detector theory

The fundamental equation for the scattering of light from polymer solutions is the Zimm equation.

M is the molecular weight of the polymer sample and C is the sample concentration. A₂ is the second virial coefficient of the solution, which corrects for the interaction of polymer molecules with each other. A₂ may be calculated from the concentration dependence of the light scattering signal. R_θ is the excess Rayleigh scattering ratio of the solution above that of the pure solvent, measured at angle θ with respect to the incident beam.

P_θ is exactly equal to unity for all molecules when θ is zero.

P_θ is 0.98 for molecules with radius of gyration (RG) of 150 nm when θ is 7 degrees.

R_G of 150 nm constitutes the upper limit of separation of GPC.

Therefore, with LALS, multiple angle measurements are unnecessary because extrapolation or correction for angular dissymmetry is unnecessary. This reduction to a single angle greatly simplifies the processing of data from multiple detectors. K is a composite of optical and fundamental constants.

n₀ is the refractive index of the solvent.
v is the refractive index increment of the polymer solution.
N_A is Avogadro’s number.
λ₀ is the wavelength of the incident light in vacuum.
p is an integer equal to 2 for vertically polarised incident light, 1 for un-polarised.

The latter three parameters (N_A, λ₀ and p) are instrument constants and can be merged with the detector constants k and I₀ in equation 2 to form a new constant which we will call Lals.Cal. Equation 1 can now be rearranged to separate the variables of molecular weight and concentration from the other parameters.

At normal GPC/SEC concentrations the second virial coefficient A₂ is typically insignificant compared to the first term in the denominator, so equation 4 can be simplified by neglecting the A₂ term.

Equation 4 is the general equation but equation 5 will be used in the present discussion for purposes of simplification.

Elution Profiles

The elution profile consists of successive fractions of the eluent sampled at equally spaced time intervals, i. Each fraction will be characterised by its molecular weight Mi and concentration C_i. The term C_i is determined from the following equation, derived in the refractometer.

RI_i is the signal from the RI detector at interval i and RI.Cal is the detector calibration constant. Substituting equation 6 into equation 5 yields the elution profile of molecular weight.

Note that the array index for the LALS signal is offset from that of the RI detector by an amount δ. This detector offset reflects the fact that the two detectors do not measure each fraction simultaneously. They have a time (volume) separation between them corresponding to the volumes of the two detectors plus the volume of the interconnecting tubing. Equation 7 reveals that molecular weight elution profile is proportional to the ratio of the LALS detector signal to the RI detector signal.

An elution profile for molecular weight on a broad distribution polystyrene sample is shown in Figure 2, overlaid with the RI and LALS signals. Notice that the Mi profile has more noise at each end than in the middle. This is due to the RI and LALS signals being lower in magnitude near the ends of the peaks, so the noise is relatively higher. In fact, the calculation of M_i becomes so unreliable on the ends that the calculation must be truncated at some point and M_i is obtained for the rest of the distribution by extrapolation. The extrapolation is shown in Figure 2 as the dashed line and is much more extensive on the low molecular weight end.

Determination of instrument constants (calibration)

The instrument constants RI.Cal, Lals.Cal and δ are determined from the chromatograms of a narrow distribution polymer standard - for example, polystyrene standard 0K, shown in Figure 3. The offset δ is easily determined as the difference in the peak positions. Figure 4 shows the same chromatograms after the offset is applied. The RI calibration constant is determined from the RI peak area and dn/dc as shown previously in Part 1. The LALS calibration constant is determined from the LALS peak area, which is directly proportional to the weight-average molecular weight as shown in the next section.

Molecular Weight Distribution

The molecular weight distribution can be represented in several ways - the most important being the number-average molecular weight M_N and the weight average molecular weight M_W. M_N is defined as the average molecular weight (molar mass) over the successive fractions of the sample, with the statistical weight of each fraction being the number of molecules, or molar concentration N_i. The molar concentration is simply the ratio of the weight concentration C_i and molar mass M_i, which are determined by equations 6 and 7, respectively.

The number-average molecular weight is therefore defined as follows:

The weight-average molecular weight is defined as the average molecular weight (molar mass) over the successive fractions of the sample, with the statistical weight of each fraction being the mass of molecules, or weight concentration, C_i.

The summations in equations 9 and 10 are directly related to peak areas in the GPC chromatogram. The sum of C_i is proportional to the area of the RI detector peak, per equation 6.

The sum of C_iM_i is proportional to the area of the LALS detector peak, per equation 5.

Combining equations 10-12 reveals a very simple relationship between M_W and the chromatographic peak areas.

The sum of C_i/M_i is proportional to the area of the molar concentration peak, per equation 8. So M_N can also be considered a ratio of peak areas as follows:

However, molar concentration is a derived function, not a detector signal, so M_N does not have the same type of simple relationship to detector peak areas as does M_W. The derivation of molar concentration often results in greater error for determination of M_N than for M_W. The reason for this is illustrated in Figure 5 where the relevant peaks are overlaid for a broad distribution sample, polystyrene in THF. M_W can be determined very precisely because it is simply proportional to the ratio of the LALS and RI peak areas, both of which have excellent signal/noise. M_N is determined by the peak area of molar concentration, which has considerable noise on the long elution side of the peak. This noise is inherent in a broad distribution sample and arises from the fact that molar concentrations are higher where the molar mass (and therefore the light scattering signal) is lower. M_N will therefore be determined by light scattering with less precision than MW where the distribution is broad. Narrow distribution samples will not exhibit this problem.

In summary, the LALS detector provides sensitive, simple and accurate molecular weight measurement for GPC/SEC. By measuring directly at very low angle it avoids the complexity, assumptions and errors inherent in angular fitting functions and angular correction.

Systems for GPC/SEC:

		Viscotek TDAmax - Integrated, temperature controlled GPC/SEC system The Viscotek TDAmax is a complete, temperature controlled, advanced, multidetector GPC/SEC system suitable for all macromolecular applications, particularly research.  It consists of three unique and complementary components – The Triple or Tetra Detector Array (TDA), the GPCmax integrated solvent and sample delivery module and the OmniSEC software.
		Viscotek 270max - Modular, ambient temperature GPC/SEC system The Viscotek 270max is a modular advanced multi detector detector system that operates at ambient temperature.  It is perfect for the routine full characterization of natural and synthetic polymers, copolymers and proteins.
		Viscotek RImax - Modular, conventional GPC/SEC system The Viscotek RImax is a modular, conventional calibration system.  It offers simple operation and full upgradeability to advanced detection.  Designed for routine GPC/SEC and teaching purposes.  Operates with the same powerful OmniSEC software as used in the advanced systems.