Published on: **Mar 3, 2016**

- 1. 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 the axes. 2. We use the 3 Dimensional (Multivariate) Calculus to find the total surface area of the mesh. 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 2. 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
- 2. 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 2. 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 (m-1)x(n-1) 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. 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 unit-less. 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.