In thomasgredig/nanoscopeAFM: Load and Analyze Atomic Force Microscopy Data Files

knitr::opts_chunk$set(
  collapse = TRUE,
  fig.width = 6,
  comment = "#>"
)

library(nanoscopeAFM)
library(ggplot2)

The Height-Height Correlation Function is discussed in the publication: Height-Height Correlation Function to Determine Grain Size in Iron Phthalocyanine Thin Films Thomas Gredig, Evan A. Silverstein, Matthew P Byrne, Journal: J of Phys: Conf. Ser. Vol 417, p. 012069 (2013).

It defines a correlation function $g(r)$ as follows:

$$ g(r) = <| (h(\vec{x}) - h(\vec{x}-\vec{r}) |^2 > $$

where $h(\vec{x})$ is the height at a particular location. The height difference is computed for all possible height pairs that are separated by a distance $|\vec{r}|$. Immediately, if follows that $g(0)=0$.

For a set of images, the correlation function can be interpreted as follows:

$$ g'(r) = 2\sigma^2 \left[ 1- e^{- \left(\frac{r}{\xi}\right)^{2\alpha}} \right] $$

which has 3 fitting parameters, the long-range roughness $\sigma$, the short-range roughness $\alpha$ (Hurst parameter), and the correlation length $\xi$.

Example: HHCF

The computation is implemented in this package as follows, the image is flattened, then HHCF is computed and the curve is fit. Choosing a large number of iterations is crucial. The computation is executed with the AFM.hhcf() function.

filename = AFM.getSampleImages(type='tiff')
a = AFM.import(filename)
a = AFM.flatten(a)
AFM.hhcf(a, numIterations= 3e5)

The correlation function data can be further analyzed, after exporting the data to a data frame:

hhcf.DF = AFM.hhcf(a, numIterations= 2e5, dataOnly = TRUE)

ggplot(rbind(hhcf.DF,c(0,0,0)),
       aes(r.nm, g )) + 
  geom_smooth(size=1, col='pink') + 
  geom_point(size=4, col='black') +
  geom_point(size=3, col='red') +
  xlab('r (nm)') + 
  ylab(expression(paste('g (nm'^2,')'))) + 
  scale_x_continuous(limits = c(0,150)) +
  theme_bw()