indentationTools.jl

IndentationTools.jl is a package of help functions to process indentation experiments. It is particularly adapt at handling AFM-based indentation data. A switch is included to allow the import of force-indentation curves directly. Currently, this option is mostly used to compare against commercial software.

This repository is written with a specific work-flow in mind. Before diving into your analysis, take a day to read the code and check that used logic applies to your problem.

Essentials

The effect of a non-rigid indenter can be accounted for by defining a measured modulus $E_r$ via

$$\frac{1}{E_r} = \frac{1-\nu^2}{E} + \frac{1-\nu_i^2}{E_i}$$

where $E$ and $\nu$ are the properties of the tested material and $E_i$, $\nu_i$ are the properties of the indenter. The key underpinning of all the equations in the package is that the measured modulus $E_r$ is related to the unloading stiffness $S$ and the projected area of elastic contact $A$ by the equation

$$ S = \frac{dP}{dh} = \frac{2}{\sqrt{\pi}} E_r \sqrt{A}$$

Unloading curve fitting equations

Oliver-Pharr [1]

The classical method from reference [1] below. Assumes that the unloading curve can be described by the equation

$$P(h) = B(h-h_f)^m $$

where $B$, $h_f$ and $m$ are fitting parameters. The initial unloading slope $S$ is found by differentiating this expression and evaluating it at the onset of unloading,

$$ S = mB(h_{\textrm{max}}-h_f)^{m-1}$$

Note that the value of $m$ predicted by theory is 1 / 1.5 / 2.0 for a flat/paraboloid/conical indenter, respectively (p. 1575, [1]).

Where can I read more about the equations implemented?

The basics of indentation testing is the theory of Hertzian contact. Beyond a general understanding of Hertz contact theory, I recommend the following articles to understand this repository:

[1] Oliver, W. C., & Pharr, G. M. (1992). 
An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. 
Journal of Materials Research, 7(6), 1564–1583. [https://doi.org/10.1557/JMR.1992.1564](https://doi.org/10.1557/JMR.1992.1564)

[2] Oliver, W. C., & Pharr, G. M. (2004). 
Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology. 
Journal of Materials Research, 19(1), 3–20. [https://doi.org/10.1557/jmr.2004.19.1.3](https://doi.org/10.1557/jmr.2004.19.1.3) 

[3] Feng, G., & Ngan, A. H. W. (2002). 
Effects of creep and thermal drift on modulus measurement using depth-sensing indentation. 
Journal of Materials Research, 17(3), 660–668. [https://doi.org/10.1557/JMR.2002.0094](https://doi.org/10.1557/JMR.2002.0094) 

[4] Tang, B., & Ngan, A. H. W. (2003). 
Accurate measurement of tip - Sample contact size during nanoindentation of viscoelastic materials.
Journal of Materials Research, 18(5), 1141–1148. [https://doi.org/10.1557/JMR.2003.0156](https://doi.org/10.1557/JMR.2003.0156) 

[5] Cheng, Y. T., & Cheng, C. M. (2005). 
Relationships between initial unloading slope, contact depth, and mechanical properties for conical indentation in linear viscoelastic solids. 
Journal of Materials Research, 20(4), 1046–1053. [https://doi.org/10.1557/JMR.2005.0141](https://doi.org/10.1557/JMR.2005.0141) 

[6] Cheng, Y. T., & Cheng, C. M. (2005). 
Relationships between initial unloading slope, contact depth, and mechanical properties for spherical indentation in linear viscoelastic solids. 
Materials Science and Engineering A, 409(1–2), 93–99. [https://doi.org/10.1016/j.msea.2005.05.118](https://doi.org/10.1016/j.msea.2005.05.118) 

Understanding the code structure

The code can be divided into X steps:

1. Data import

The data import is done by reading the file that was exported from the AFM machine. Helper functions are provided for certain file formats we worked with in the past, such as .IBW files.

The general sequence of events is as in the snippet below where a CSV file containing 5 columns of numerical data and 6 rows of headers is imported. Note that it is assumed that $xy$ always contains the indentation depth in column 1 and the indentation force in column 2.

Although the units are arbitrary as long as they are consistent, we found it convenient to work with nano-meters, nano-Newtons and nano-meters squared.

Note that in the example below, the variable "filename" is supplied by you and is the path to the file with data exported from the AFM machine.

# Dependencies (only necessary if writing your own importer)
using CSV   
using Plots
using DataFrames

# Read file (check CSV documentation for the different keywords)
df = CSV.File(filename, decimal = '.', skipto = 7, delim = "\t", header = ["Depth_nm", "Load_uN", "Time_s","Depth_V","Load_V"]) |> DataFrame

# Throw away columns that are not needed
df = deepcopy(df[:,1:2])

# Convert the dataframe to a matrix (df should contain only numeric data)
dx = Matrix(df)

# Convert force to nano-Newtons
xy[:,2] *= 1.0e3

# Convert to Float32
xy = Float32.(xy)

2. Data preprocessing

Preprocessing consists of a number of steps with the end goal of isolating the unloading part of the experiment. In practice, this means that Step 2 can be further decomposed into:

• 2a. Finding the point where the indentor comes into contact with the substrate. This determines the (0,0) point in the experiment. The function offsetAndDriftCompensation can often quite relibably determine this point. Note that offsetAndDriftCompensation overwrites the $xy$ matrix.

  # Find initial contact
  xy , ~ , ~ , rampStartIdx, ~  = offsetAndDriftCompensation(xy)
  
  # Plot
  ctrl.plotMode && display(plot([xy[:,1]],[xy[:,2]]))

• 2b. Finding the start of the hold. In all our experiments, the hold occurs at maximum load or maximum displacement.

  Note that hyperParameters is a structure supplied by you. You can check the values that hyperParameters should contain by typing ?hyperParameters when you have indentationTools.jl loaded.

  # Determine the start of the hold time at circa max force.
  if cmp( lowercase(hyperParameters.controlLoop) , "force") == 0
      holdStartIdx = findStartOfHold(xy,"first")
  elseif cmp( lowercase(hyperParameters.controlLoop) , "displacement") == 0
      holdStartIdx = argmax(xy[:,2])
  else
      throw(DomainError( string(hyperParameters.controlLoop), "controlLoop setting not defined."))
  end

3. Fitting of the slope