Approach for differentiating nanoscale materials based on the size criterion using number of modes by count and size, locations of modes by size, mean size count, and standard deviations centered around mean sized count. 
Background
EPA is proposing in its proposed 8(a) and significant new use rules to differentiate discrete nanoscale materials (NMs) based on the numeric middle of a particle size range whose resolution varies according to the width of the range (the current proposal outlines five ranges: 1  -  20 nm; 20  -  50 nm; 50  -  1000 nm; and 1000  -  10000 nm).  For example, the numeric middle of a NM with 90% of the primary particle sizes falling between 55 - 65 nm is 60 nm.  There are several considerations and parameters to consider beyond this specific approach:
   1. No scientific literature has differentiated or identified NMs based on the numeric middle of the size range. 
   2. Extensive scientific and mathematical literature has dealt with distributions of various properties and typically identifies four properties to characterize a distribution, as well as the type of distribution itself: mean or average, variance, skewness, and kurtosis.   Common distributions include: normal, Poisson, Rayleigh, exponential, and geometric, but there are many others. 
   3. Figures 1-3 illustrate different particle size distributions that, under the proposed approach, would be considered to be the same "nanoscale material" despite scientific studies demonstrating that these substances have different environmental properties (Auffan, Rose et al. 2009):
         a. Figure 1 

Figure 1 illustrates a broad distribution and a narrow one that have the same "middle" of the range.  The peaks are also constructed to have the same mean of particles by count, though not by mass.  The black line indicates the mean at 50nm, which is also the middle of each range, while each dashed vertical line represents two standard deviations from the mean (roughly 95% of the particle size distribution) for the respective distributions. 

Figure 2 illustrates two materials have different means and skewness (there is >5nm difference between the means), but 95% of the particle s fall into the same size range (represented by the dashed vertical lines) . The distribution for material 1 has relatively small particles and is skewed left, while the distribution for material 2 has relatively large particles and is skewed right.  Even though the majority of the particle sizes are different in each distribution, they would still be considered the same "material" under the current proposal.  The mean by count for each material are represented by the solid vertical lines. 

Figure 3 illustrates a material with bimodal distribution (#1) with the same mean and width of range as material #2.  Line a represents the mean for both substances, which also happens to be the median for material #2.  The median for material #1 is at line b.  Lines c & d are the modes for material #1. 

Although the proposed approach as described in the current rule could simplify understanding of and compliance with the rule, the differences in environmental parameters based on different size distributions support the idea for a more rigorous approach.  At the same time, more complexity or nuance does not mean a regulated entity would be unable to meet the requirements, because the proposed revised approach is based on statistical distributions science. 
Foundations for an alternative approach
Although the mathematics of distributions has been extensively described, a rigorous differentiation of NMs based on particle size distributions utilizing a particular type of distribution or distributions, such as utilizing Student's T-test, is probably not warranted. There are several reasons for this:
   1. There are several different types of "particle" including, but not limited to:
         a. Primary particle
         b. Grain
         c. Aggregate
         d. Agglomerate
No single method or protocol exists for characterizing these properties of a single material that can span orders of magnitude of size or to adjust for different conditions.   

   2. Under the Nanoscale Material Stewardship Program and the nanoscale notices received under the New Chemicals Program, the Agency has frequently seen particle size distributions that are bimodal size distributions typically representing aggregation and agglomeration processes.  There are several other processes that might generate a bi- or trimodal distribution including particle formation processes that occur under threshold reaction conditions (this might also lead to graininess).  There may be others.

   3. Different shapes will also lead to issues of defining the boundaries of distributions of critical parameters for non-spheroid nanoscale materials like fibers, trees, etc. 

   4. Utilizing statistical tools, such as a Student's t-test, to identify whether a particular size distribution is the same or different than another size distribution is a scientifically objective approach to settling the question as to whether two materials are the same from a size perspective within specified limits of statistical uncertainty.  However, this would require embedding the distribution information into the nomenclature of a discrete material.  This strict approach is well outside the norm of how EPA considers distributions in Unknown Variable Composition and Biological (UVCB) substances for chemical nomenclature.  Second, it would place a large burden on industry to submit and EPA to assess potentially new submissions. 
Taking into account the above issues, and given the need to balance simplicity and taking a case-by-case approach, the following framework is an alternative proposal for differentiating NMs from each other based on size.  The other parameters outlined in the proposed rule would remain the same.  
Alternative approach

    1. The key parameters of size distributions are:
          a. Number of modes by count and size
          b. Locations of modes by size
          c. Mean size by count
          d. Two standard deviations centered around mean size by count
             
    2. Each chemical substance that meets the definition of a nanoscale material should be evaluated as to the number of type of critical dimensions.  
       
Scale of dimensions (a, b, and c)
Example Shapes
Number of critical dimensions
Dimensions
Notes
abc

Spheroid, cubes
1
Diameter
Zingg's shape: equant
abc
Irregular shapes, cones
1
Equivalent spherical diameter
Submitter should convert volume or surface area of NM into an equivalent spherical diameter

ab<c 
Fiber, needle, rod, 
2
Length and diameter
ab should be converted to equivalent circular diameter
Zingg's shape: prolate
ab>c
Ring, sheet, torus
2
Depth and diameter
ab should be converted to equivalent circular diameter
Zingg's shape: oblate
a<b<c
Ribbons, attached or unattached
3
Length, width, depth
Zingg's shape: bladed
       Two dimensions should be considered equivalent or on the "same scale" when each dimension's mean by count are within 3/2 of each other.  This approach was developed by Zingg for use in describing 3-dimensional objects in geology (Zingg 1935)}.  Materials that are "hollow" or that encapsulate another substance should have another critical dimension: internal volume.  
       
    3. Each critical dimension should be defined by the four parameters of the size distribution as follows:
       
          a. Number of modes by integer number provided the distribution is statistically relevant.  Hence, a bimodal distribution is different than unimodal distribution. A mixture of two normal distributions with equal standard deviations is bimodal only if their means differ by at least twice the common standard deviation (Schilling, Watkins et al. 2002)
          b. Locations of modes by size, mean size by count, two standard deviations centered around mean size by count by the following approaches to size resolution:
                 i. for parameters up to 20 nm, the parameter differs from that of other materials by 2 nm or more,
                 ii. for parameters between 20 and 50 nm, the parameter differs from that of other materials by 5 nm or more,
                 iii. for parameters between 50 and 1000 nm, the parameter differs from that of other materials by 10 nm or more,
                 iv. for parameters between 1000 and 10000 nm, the parameter differs from that of other materials by 100 nm or more,
                 v. for parameters greater than 10000 nm, the parameter differs from that of other materials by 1000 nm or more
                   
    4. Nanoscale materials shall be distinguished by distributions at the primary, secondary, and tertiary structural level, e.g. primary particles, aggregates, and agglomerates.  Hence, an aggregate with a mean size by count of 8 microns would be different than an aggregate with a mean size by count of 4 microns, even if  the nanoscale primary particles of each substance would be considered to have the same size distribution.  
       
Auffan, M., J. Rose, et al. (2009). "Towards a definition of inorganic nanoparticles from an environmental, health and safety perspective." Nat Nano advance online publication.
Schilling, M. F., A. E. Watkins, et al. (2002). "Is Human Height Bimodal?" The American Statistician 56(3): 223-229.
Zingg, T. (1935). "Beitrag zur Schotteranalyse." Schweizerische Mineralogische und Petrographische Mitteilungen 15: 39 - 140.

       
