DLS reports 65 nm but TEM shows 20 nm for the same nanoparticles: which size is correct?

Neither is wrong; they measure different things. TEM shows the dry, electron-dense core. A polymer coating has little contrast and collapses in the vacuum, so it is mostly invisible. DLS measures how fast the particles diffuse in the liquid and converts that to the diameter of a hard sphere that would diffuse at the same rate, the hydrodynamic diameter. That includes the core, the swollen polymer shell and the layer of solvent that moves with the particle.

So a hydrodynamic size above the TEM size is expected for coated particles. A shell adding 10 nm on each side would take a 20 nm core to 40 nm. Reaching 65 nm suggests something more, usually a fraction of clusters.

The conversion DLS uses is the Stokes-Einstein relation: d_H = k × T / (3π × η × D), where D is the measured diffusion coefficient and η the viscosity of the liquid. As a reference, a 100 nm sphere in water at 25 °C (η = 0.89 mPa·s) has D = 1.38 × 10^-23 × 298 / (3π × 0.89 × 10^-3 × 10^-7) = about 4.9 × 10^-12 m²/s.

The result depends directly on the temperature and viscosity entered in the software. Measuring in a buffer, a glycerol mixture or a concentrated polymer solution while leaving the viscosity set to pure water gives a wrong size with no warning.

The weighting is the larger effect. For particles much smaller than the wavelength, scattered intensity rises with the sixth power of the diameter. One 100 nm cluster scatters as much light as a million 10 nm particles. The Z-average is an intensity-weighted mean, so a small number of aggregates pulls it far above the size of the majority, while TEM gives a number-weighted distribution where those few aggregates barely count.

Your polydispersity index of 0.35 supports this. Values below about 0.1 indicate a narrow, single population; at 0.35 the sample is broad or contains more than one population, and the single Z-average figure is a poor description of it.

To find out what is really in the suspension: look at the intensity distribution rather than only the Z-average, and see whether there is a second peak at a few hundred nanometres. Filter the sample through a 0.2 µm syringe filter, or centrifuge gently, and measure again; if the Z-average drops sharply, dust or a few large aggregates were responsible. Measure at two or three dilutions to rule out multiple scattering and particle interactions.

If the software offers a number-weighted distribution, use it only as a rough guide. It is calculated from the intensity data with assumptions about the optical properties, and small errors are amplified enormously by the sixth-power relation.

For reporting, give both and say which is which: the core diameter from TEM as a mean and standard deviation from a few hundred counted particles, and the hydrodynamic diameter from DLS as Z-average with the polydispersity index, stating the dispersant, temperature and concentration. They answer different questions. Core size governs the magnetic properties of iron oxide, while hydrodynamic size governs how the particles diffuse, sediment and pass through filters or membranes.

Remember that TEM has its own bias. Drying on the grid can pull particles together, so clusters in an image are not proof of aggregation in the liquid, and a few images cover only a tiny number of particles.