Size: What is the z-average?

The Z average is the intensity weighted mean hydrodynamic size of the ensemble collection of particles measured by dynamic light scattering (DLS). The Z average is derived from a Cumulants analysis of the measured correlation curve, wherein a single particle size is assumed and a single exponential fit is applied to the autocorrelation function.

The DLS autocorrelation function, along with the exponential fitting expression, is shown below, where I is the scattering intensity, t is the initial time, τis the delay time, A is the amplitude or intercept of the correlation function, B is the baseline, D is the diffusion coefficient, q is the scattering vector, λis the vacuum laser wavelength, ñ is the medium refractive index, θ is the scattering angle, k is the Boltzmann constant, T is the absolute temperature, η is the viscosity of the medium, and Ris the hydrodynamic radius.

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In the Cumulants approach, the exponential fitting expression is expanded to account for polydispersity or peak broadening effects, as shown below.

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The expression is then linearized and the data fit to the form shown below, where the D subscript notation is used to indicate diameter. The 1st Cumulant or moment (a1) is used to calculate the intensity weighted Z average mean size and the 2nd moment (a2) is used to calculate a parameter defined as the polydispersity index (PdI).

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It is important to note here, that the Cumulants analysis algorithm does not yield a distribution – it gives only the intensity weighted Z average and the polydispersity index.

The figure below shows a graphical representation of the procedures employed within the Cumulants analysis to calculate the Z average and polydispersity index.

The initial slope (blue) of the correlation function is related to the z-average size, the curvature (red) indicates the nonlinearity due to the width of the polydispersity.

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