Nano Scale Roughness Quantification
Published on: Mar 3, 2016
Transcripts - Nano Scale Roughness Quantification
To: Kelly Prior, Vice President of Research, Ligoure Labs
From: Team 3
Subject: Re: Procedure for quantifying the roughness of Diamond Samples
Date: February 27, 2014
Our team was asked to create a procedure. The direct user is the Materials Research
Team. The direct user needs a deliverable that provides a way to quantify the roughness of new
nano-scale coatings. To quantify the roughness of nanoscale surfaces, the criteria for success of
this deliverable are quick, easy-to-use, and able to quantify the roughness of diamond
samples. The constraints are that we are provided with limited amount of data to test our
solution and that the solution must consume less time in calculation (since we use a large data
set). Our procedure involves comparing (which involves finding the ratio of) the surface area of
the nano-sample to that of the base area.
An assumption we consider is that the procedure will work for diamond samples given
that it is used on the provided gold samples. Another assumption is that our procedure will be run
under consistent conditions (the temperature and pressure do not vary the data over time). The
limitation of our procedure is approximately above 95% accuracy to solution at the nano-scale
and as the scale/size of data decreases the solution starts to lose accuracy.
Our definition of roughness of is the amount of deviation a nano-surface has from its base
area (totally flat area) because the rougher the surface is, the more surface area it has. This means
that we are going to compare (by finding the ratio of) the surface area of the nano-sample to that
of the base area which is our reference level (area of the two dimensional plane beneath the
surface). We will call this ratio roughness factor which will essentially quantify the roughness.
The higher the roughness factor, the higher the roughness.
The procedural steps to determine roughness of nano-scale surface are:
1. For the calculations, disregard/remove the first row and first column as they are x and y
axes. The rationale for this is to allow the direct user to reduce the data into manageable
parts and make the calculations easier. This results in the base area to be simply measured
as just rows multiplied by columns. Due to the nature of the end result (the roughness
factor) being a ratio, the result of any data tested will be equally affected by not including
2. We use the 3 Dimensional (Multivariate) Calculus to find the total surface area of the
3. The formula we use is the integration of √(1+fx
2 + fy
2) where fx is partial change in
height when y is kept constant and fy is partial change in height when x is kept constant.
We approximate this formula to use to our data.
4. The data has m rows and n columns. Starting from the first row, take each element of each
row and subtract it from the corresponding element in the following row. Follow this pattern
until the (m-1) row. Thus we get (m-1) horizontal vectors each having n elements. This
matrix formed is fx for each point of data and has dimensions (m-1)x(n). Square all the
elements in the obtained matrix. This is fx
5. Now we do the same with n columns. Starting from the first column, take each element of
each column and subtract it from the corresponding element in the following column. Follow
this pattern until the (n-1) column. Thus we get (n-1) vertical vectors each of size m. The
matrix formed is fy for each point and has dimensions (m)x(n-1). Square all the elements
in the obtained matrix. This is fy
6. It is noted that fx
2 and fy
2 have dimensions (m-1)x(n) and (m)x(n-1) respectively. In
order to make their dimensions agree, we remove the last column of the fx
2 matrix and last
row of fy
2 matrix. Now both fx
2 and fy
2 have the same dimensions (m-1)x(n-1).
7. We now add fx
2 and fy
2. We obtain (fx
2 + fy
2) at every point on data. To every element
of this sum, we add 1 to obtain (fx
2 + fy
2 + 1) at every point. Note that it is of dimensions
8. Now we take the square root of every element to obtain √(fx
2 + fy
2 + 1) which is
equivalent to the surface area of every data point.
9. We add all these data points to obtain the total surface area of the sample.
10. We now calculate the Base Area by multiplying m (the number of rows) by n (the
number of columns).
11. Now we divide the total area by the base area which gives us a quantity we defined as
Roughness Factor = Surface Area / Base Area
Roughness Factor of A = 1.36x106/ 5.18x104 = 26.17
Roughness Factor of B = 7.78x105/2.59x104 = 29.98
Roughness Factor of C = 1.48x105/2.59x104 = 5.71
The rationale for our models critical steps of calculating the approximate surface area
(using all the three coordinates of the 3D surface) are based on the fact that a more rough surface
has a larger surface area than a flat completely smooth base area. Of the data provided, our
procedure uses the height, location, and its relation to other data points for each piece of data in
the spreadsheet data sets. Note that our roughness factor has no units because both surface and
base area have same units and the ratio of roughness factor cancels out these units making it
Our team has identified the complexity of this problem as how to figure out how to use
and sort through a large amount of provided data and that our solution must be effective but
simple enough for the research team to use.
Our procedure addresses the problem’s complexity by creating a mathematical model that
can be applied to any set of data. Our model deals with the different features (surface, partial
changes) of the AFM image provided by calculating the approximate surface area of the three
dimensional graph. We consider the image size by considering the dimensions of the matrix data
provided. These are the roughness factors that we obtained:
Sample A : 26.17 Sample B : 29.98 Sample C : 5.71
Thank you for the opportunity to design a procedure to solve this important problem. We
hope this procedure helps the Materials Research Team to test the smoothness of surfaces.