Calculating the Contour Length and Hydrodynamic Radius of Polyethylene Glycol in Aqueous Solutions

Polyethylene glycol (PEG), a versatile hydrophilic polymer, is widely employed in biomedical applications such as drug delivery and protein conjugation due to its biocompatibility and tunable chain dimensions. Understanding the structural parameters of PEG chains in water is essential for predicting their behavior in solution, including diffusion, steric shielding, and interactions with biological interfaces. This post outlines the fundamental methods for calculating the contour length—the maximum extended chain dimension—and the hydrodynamic radius, which characterizes the effective size of the solvated coil. These calculations rely on established polymer physics principles and empirical relations derived from experimental data.

Contour Length: The Fully Extended Chain Dimension

The contour length (Lc​) represents the total length of the PEG chain if it were fully stretched in an all-trans conformation, serving as an upper bound for its spatial extent. PEG consists of repeating ethylene oxide (EO) units (–CH₂CH₂O–), each with a molecular weight of approximately 44 g/mol.

To compute Lc​, first determine the number of EO units (n) from the polymer’s molecular weight (MW) in g/mol: ​

The projected length per EO unit in the extended state is typically 0.35 nm, accounting for the zig-zag backbone geometry. This value arises from the C–C and C–O bond lengths (≈0.154 nm) and bond angles, yielding a repeat distance of about 3.5 Å.

Thus, the contour length is:

Example: For PEG with MW = 5,000 g/mol, n≈114, so Lc≈114×0.35=39.9 nm. This metric is solvent-independent but highlights the chain’s potential reach, contrasting with its coiled state in solution.

Note: there are different opinions on the fully extended PEG length in water. It is estimated that each EO unit has a length around 0.28 nm in water per this paper Single molecule force spectroscopy by AFM
indicates helical structure of poly(ethylene-glycol) in water in New Journal of Physics 1 (1999) 6.1–6.11. This estimate is about 20% shorter than the above mentioned 0.35 nm length published in this paper in Polymer Journal volume 47, pages464–467 (2015).

Hydrodynamic Radius: The Effective Size in Water

In aqueous environments, PEG adopts a flexible random coil conformation, solvated by 2–3 water molecules per EO unit, which expands its effective volume. The hydrodynamic radius (Rh​) quantifies this solvated size, modeling the coil as an equivalent sphere that diffuses at the same rate as the polymer. It is experimentally derived from techniques like dynamic light scattering or viscosity measurements via the Stokes–Einstein relation:

where kB​ is Boltzmann’s constant, T is temperature, η is solvent viscosity, and D is the diffusion coefficient. For practical estimation, empirical power-law scaling is used:

where v≈0.55 reflects near-ideal chain behavior in water (Flory exponent for good solvents), and a≈0.023 (with MW in g/mol) fits data for MW up to 35 kDa. A more precise form from viscosity studies is R in nm for medium MW ranges according to papers 1 and paper 2.

Example: For the same 5,000 g/mol PEG, Rh≈0.023⋅(5000)^0.55≈2.3 nm. This is roughly 1/17th of the contour length, underscoring the coil’s compactness, with the full effective hydration diameter ≈4.6 nm.

Example: For linear PEG 20,000 g/mol, its hydrodynamic diameter is about 12 nm.

Paper 1: Hydrodynamic Radii of Polyethylene Glycols in Different Solvents Determined from Viscosity Measurements, J. Chem. Eng. Data 2008, 53, 1, 63–65, https://pubs.acs.org/doi/10.1021/je700355n

paper 2: Molecular Dynamics Studies of Polyethylene Oxide and Polyethylene Glycol: Hydrodynamic Radius and Shape Anisotropy, Biophysical Journal Volume 95 August 2008 1590–1599, https://doi.org/10.1529/biophysj.108.133025