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Particle Size/Size Distributions NanoPlus Operator’s Manual
E-6 Mar 2014
Particle Size/Size Distributions
The Cumulants Method
For monodisperse particles, it can be seen from equation (4) that the logarithm of G
1
(

becomes a
straight line. For polydisperse samples, the logarithm of G
1
(

exhibits a curved line.
In the Cumulants method, one fits the logarithm of G
1
(

to a polynominal in

to determine the
coefficients, Km:
(8)
The first order coefficient (or the slope of lnG
1

is the average decay constant,

, from which the
average diffusion coefficient and particle diameter can be calculated using equations (5) and (1).
The second order coefficient divided by the square of

is the polydispersity index, ( <(

-<

> ) 2 >
/ <

> )
2
). The value of the polydispersity index is low, typically smaller than 0.1, for
monodisperse samples and becomes larger for polydisperse samples.
Particle Size Distribution Methods
There are three methods for particle size distributions from the measured autocorrelation
functions.
The CONTIN Method
The well-documented computation routine known as CONTIN utilizes the regularized
nonnegative least-squares technique combined with the Eigenfunction analysis. It has been
implemented as the main data retrieval algorithm. CONTIN uses a non-linear statistical technique
to smooth the solution and reduce the number of degrees of freedom to an acceptable level. The
User is required to specify the distribution range, the number of data points, and other constants.
The regularization parameter can be automatically chosen. CONTIN takes into account the
weighting of the distribution due to the use of discrete data points in the continuous distribution
and then calculates different moments of the computed distribution.
The Marquardt Method
The Marquardt method is an iterative method in which an initial guess is repeatedly processed to
give a final answer. In the analysis program, the initial guess is a histogram with all the steps being
of equal height. The algorithm is repeated to change the histogram to fit the raw data. The number
of times the algorithm is repeated is the step number. The lambda parameter adjusts how much the
answer changes in one step. If it is small, the answer changes rapidly, and if large, the answer
changes slowly. Distributions obtained by this method tend to be broad and have connected peaks.
G
1
  
ln
K

–
 
m
m
-------------------
 
 

=