# Rh or Rg: Which is best for my GPC/SEC samples?

The primary goal when employing gel permeation / size exclusion chromatography (GPC/SEC) is often to determine the molecular weight of a macromolecular sample. Multi-detector systems, such as Malvern Panalytical’s OMNISEC and TDAmax systems, are able to offer a range of other molecular characterization parameters, including molecular size. In some situations, such as the analysis of proteins or other samples where the molecular weight is already known, the sole purpose of the analysis is to determine the molecular size of the sample. In these instances, the molecular size can reveal critical information about the sample’s molecular structure, which is often tied closely to the sample’s ultimate function or application.

There are two common measurements of molecular size available through GPC/SEC analysis: hydrodynamic radius (Rh) and radius of gyration (Rg). A previous post describes how Rh and Rg are utilized in protein characterization using dynamic light scattering (DLS) and small-angle X-ray scattering (SAXS). This post will discuss the definitions of Rh and Rg, how the two are calculated in a GPC/SEC context, and the practical ramifications of those differences. For more details (and more equations!) on the topics herein, please see these linked documents.

**Definitions of Rh and Rg**

Without diving too deep into the math (yet), the Rh of a sample is the radius of a theoretical sphere that possesses the same mass and density that is calculated for the sample from its molecular weight and intrinsic viscosity. The Rg represents the root mean square distance of the molecule’s components from the mass center of the molecule. The single Rh and Rg numbers generated by a GPC/SEC analysis are the weight average values obtained from compiling the calculated values at each data slice. Similar to weight average molecular weight (Mw), these values are the most appropriate for characterizing the sample as a whole. It should be noted that a multi-angle light scattering (MALS) detector decoupled from a GPC/SEC instrument running in batch mode provides the z-average Rg value, Rgz.

**How Rh and Rg are calculated**

Now, we’ll get into the math just a bit. Based on Einstein’s model describing the viscosity of a solution of spherical particles, the relationship between Rh, molecular weight (M) and intrinsic viscosity ([η]) is shown below (N_{A} is Avogadro’s number).Since GPC/SEC directly calculates molecular weight and intrinsic viscosity at each data slice, the Rh can easily be determined using this equation. Therefore, the lower limit of Rh calculation aligns with the smallest sample fraction for which there is sufficient detector response in the refractive index (RI), light scattering, and viscometer channels.

To calculate Rg from GPC/SEC there are two options. The direct and most common method is to observe the change in scattered light intensity against angle of observation. This requires a light scattering detector with at least two angles to observe the sample’s angular dependence.

As mentioned in the previous discussion of Rh and Rg and a more recent post regarding light scattering detectors, it is difficult to determine Rg for proteins using GPC/SEC as proteins are generally smaller than 10-15 nm and therefore are isotropic scatterers, i.e. they scatter light equally in all directions. Low molecular weight polymers are isotropic scatterers as well and means that the resulting partial Zimm plot is flat with a slope of zero, therefore Rg cannot be determined for these smaller, materials. This is depicted on the left in Figure 1, above.

For Rg to be calculated by light scattering, the molecule must display angular dependence which means that the intensity of scattered light varies with angle of observation. This results in a sloped partial Zimm plot, illustrated by the example shown on the right in Figure 1. The wavelength of light used in the light scattering detector affects the size threshold at which a sample displays angular dependence, so it is possible that Rg can be calculated to smaller size ranges by adjusting the light source wavelength.

An alternate way to calculate Rg is by using the Flory-Fox equation, shown below, that relates molecular weight (M) and intrinsic viscosity ([η]) to Rg (Φ_{0} is treated as a constant).The advantage of calculating Rg in this manner is that, like Rh above, the lower limit of calculation depends on detector response instead of whether the sample shows angular dependence. The disadvantage is that the term Φ_{0} is actually dependent on the sample, solvent, and resulting molecular structure under study (it’s most appropriate for random coils), and thus setting it as constant introduces a level of approximation. In general, and for the remainder of this post, when Rg is discussed it is calculated from a light scattering detector.

**Differences in Rh and Rg data**

The Rh and Rg data available for a given analysis are dependent on the specific sample under investigation. Since Rg calculation requires that a molecule display angular dependence if a sample is an isotropic scatterer exhibiting no angular dependence there is a chance that Rg cannot be determined. A sample with a broad distribution of molecular weight and molecular size might possess large fractions that show angular dependence and smaller fractions that do not. In this case, Rg can only be calculated for the earlier eluting, larger molecules.

An example highlighting this behavior is shown in Figure 2. The portions of the data in which the Rh and Rg values are directly calculated are represented by the solid segments of the dark green and magenta plots, respectively. The dotted portions of the lines indicate regions where the Rh and Rg values are extrapolated. The smallest Rg value that can be calculated for the sample is 11.55 nm, which represents the point at which the later eluting fractions of the sample are too small to display angular dependence. This leaves Rg to be extrapolated for a large section of the sample. In contrast, Rh can be calculated down to 5.08 nm, allowing the molecular size of the majority of the sample to be directly calculated.

Since Rh and Rg represent separate properties of the sample, it should not be expected that the resulting values will be the same. This doesn’t mean that one of them is wrong! For a given sample type and solvent there will probably be a consistent relationship between the two. In my experience with polymeric samples (since it is difficult to obtain Rg data for proteins) the calculated Rg is on the same order but a bit larger than the determined Rh (in the example in Figure 2 Rh = 13.62 nm; Rg = 14.19 nm). This doesn’t have to be true in all cases, as the relationship between Rh and Rg is dependent on the molecular structure.

Molecular size is important to scientists for a variety of reasons. I encourage you to use the available data in the manner most appropriate for your samples. And the next time you’re presenting GPC/SEC data and someone asks about the differences between Rh and Rg, you’ll be ready!

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