# What is the maximum viscosity for DLS?

## Is there a maximum viscosity for DLS measurements? The Malvern Zetasizer can determine particle size by dynamic light scattering (DLS). In this technique, the intensity fluctuations are analyzed to find the corresponding diffusion coefficient that led to the fluctuations. The translational diffusion coefficient Dt of a particle is inversely related to its size and the viscosity of the dispersant (or more accurately the hydrodynamic radius rH and viscosity η).

This relationship can be used to predict detectability limits of the technique in general, and this is how we explore the idea of a maximum viscosity (or maximum size) border of applicability.

### Diffusion, size, and viscosity

The translational diffusion coefficient obtained from DLS is related to particle size via the Stokes Einstein equation: where the thermal energy given by the Boltzmann constant kB times absolute temperature T (in Kelvin) is divided by the viscous drag given by 6 times pi times the viscosity times the hydrodynamic radius RH. It is also occasionally seen with a factor 3, when the size is expressed as a hydrodynamic diameter instead of the radius. Since kB is constant and we are interested in measurements at room temperature for now, the above full equation can the reduce to the simplified proportionality, stating that the diffusion coefficient is inversely proportional to viscosity and size.

### How can we get the maximum size?

The specifications of the Zetasizer state that the maximum size for particles in water is 10 microns. With the help of the diffusion coefficient equation, we can now translate this to any arbitrary viscosity and predict the corresponding maximum size.

#### Approx. maximum size by DLS

Viscosity [cP]                Maximum size [nm]
1.0                   10,000
2.5                     4,000
10.0                     1,000
100.0                        100
1,000.0                          10

### What is the maximum viscosity?

The problem to solve is very similar to the maximum size. We can simply look at what the slowest diffusion coefficient for the specification at the limit is (i.e. the large size limit) and then transpose from there.

#### Approx. maximum viscosity by DLS

Size                        Maximum viscosity [cP]
10 μm                                     1 cP
1 μm                                   10 cP
100 nm                                 100 cP
50 nm                                 200 cP
10 nm                              1,000 cP

The observant reader may have noticed that we just keep the product of size and viscosity constant, so it is not too challenging to determine the combination for either a different size or a different viscosity.

### Can DLS measure viscosity?

To find the viscosity of an unknown dispersant, we may be able to use DLS to find it. In order for this method to work, we need to have some particles of known size. And we need to be certain that these particles are not interacting with the dispersant. The particles must not aggregate in the dispersant or otherwise react with it. IF we are sure that the size remains constant, then we can perform a DLS measurement of our known particles in the unknown dispersant. We compare this with the data from the particles in a known dispersant. Since DLS measures the diffusion coefficient, we can now back-calculate what the correct viscosity of the dispersant must be. Instead of calculating, you can also just edit a measurement to find the “new” viscosity.

Example: 100nm Latex beads in water measure as z-ave = 104nm. A small amount of 100nm Latex beads in the unknown dispersant (where we set the dispersant to “water”) measures as z-ave = 78nm. The unknown viscosity is then: water viscosity *78/104. You can confirm this by editing the record such that the z-ave of the edited record is 104nm.

Hope the above eliminates some confusion about the limits of dynamic light scattering.

Previously

###### If you have any questions, please email me at ulf.nobbmann@malvern.com. Thanks! While opinions expressed are generally those of the author, some parts may have been modified by our editorial team.

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