To obtain a reasonable measure of surface roughness on the nanometer scale, people most often use the atomic force microscope (AFM) or the scanning tunneling microscope (STM), not only because they offer the required resolution, but also, and more importantly, because AFM and STM images are height-encoded. This means we can measure the dimensions of the features in these images both in the plane (in x and y) and out of the plane of the sample surface (in z).

In fact, roughness my be the single most frequently made measurement in industrial applications of the AFM, and certainly an important measurement in academic research applications as well.¹

Typically, AFM users rely on root mean square (rms) roughness, S_q, as the measurement of choice. (A quick search on Google for “rms AND AFM” returned about 333,000 hits today.)²

   (Eq. 1)

where μ is the mean value of the height, z, across all in-plane coordinates (x,y):

   (Eq. 2)

This measurement, rms roughness, has some inherent limitations that are often neglected. Reporting the rms roughness is almost always useful, but frequently inadequate in accurately describing surface topography in a meaningful quantitative way. In some cases, the consequence of not knowing (or ignoring) the limitations of rms roughness is misperceptions and making poor decisions.

The limitations of rms (roughness) are well-known to those who work often and in some depth with statistics and probability theory, but not to most AFM users. The upshot of these limitations can be summarize by saying that rms roughness measurement can give nearly or identically the same numerical result for two surfaces whose roughness are qualitatively different, or very different; sometimes so different that even a simple visual inspection of the AFM images will reveal.

One important reason rms roughness is sometimes inadequate is that it is computed indiscriminately towards the polarity of the height value at a given pixel, relative to the mean height value across all the pixels in the image. In other words, as the formula (Eq. 1) shows, height values smaller than and larger than the mean value end up contributing to the rms roughness the same way, i.e., as positive numbers. The result is that the rms roughness may measure (very) nearly the same for two different surface, one for example a flat surface with many holes, the other a flat surface with many peaks.

It is clear that to distinguish at least between these two kinds of surfaces, a different kind of parameter from rms roughness is required. Such a parameter exists, and in fact it exists in most if not all commercial AFM image processing software. It is called skewness, and it is not nearly as popular with AFM users as the rms roughness is. (A quick Google search for “skewness AND AFM” returned about 15,000 hits today). The skewness, S_sk, is defined in a way that can quantitatively describe the asymmetry of a height distribution about the mean (and from there, it gets its name):³

  (Eq. 3)

The formula is similar to the one for rms, but unlike rms (roughness), the skewness can take on positive and negative values as well as zero (even if the surface is not perfectly smooth), because each term in the double summation is raised to an odd power.
Depending on the way z values are recorded in an AFM image, the mean value, μ, itself can do the same (see Eq.2), that is, take on positive or negative values. The difference is that the appearance of the third power in the double summation in Eq. 3 means that those features whose height is farther above or farther below the mean μ; make relatively heavier contributions to the computed value of the skewness, as compared to features closer to the mean.

For a symmetric surface, that is, a surface the height of whose features are statistically evenly distributed around the mean, the skewness will render a value near or equal to zero–the height distribution is not skewed.

For our example, the skewness will measure positive for the flat surface with peaks, and negative for the flat surface with holes, and yet the rms roughness may measure the same or nearly the same for both. In this example, if the sample is a piece of metal bearing whose friction performance is to be improved, then, it may make a big difference whether the surface is flat with many holes, or flat with many peaks. To settle for the rms roughness then may be to ignore the skewness risk.

I wrote earlier that the limitations of rms roughness measurement are well-known to those who work often and in some depth with probability and statistics, but not to most AFM users. In case I planted any doubt that rms roughness is overly subscribed outside AFM image analysis too, here goes: Millions of people make pretty important decisions about money using statistics and probability often, but not all of them in much depth (that‘s why I italicized the “and“). The celebrated French mathematician Benoit Mandelbrot, the inventor of fractals, has studied the implications of ignoring the skewness risk by analyzing financial markets using models (including perhaps the most famous one, the Black-Scholes model) that assume market “roughness“ has a symmetric distribution (zero skewness). Mandelbrot’s alternative ideas about this subject were published recently.⁴

Some AFM images are more pleasing to the eye than others. The beauty of every AFM image though is that it is height encoded. The availability, in the form of numerical values, of all three dimensions of every feature in an AFM image, opens up a myriad options for statistical analysis of surface topography on the nanometer scale. These options exist neither in scanning electron and transmission electron microscopy (despite their superior resolution) because the images are slope-encoded, not height encoded. Nor do they exist in other techniques that do offer height encoding, but that lack the required resolution, for example, optical and stylus profiling.

AFM (software) manufacturers are aware of the implications of this advantage beyond what the average AFM user seems to know. They implement numerous formulas in their software (and provide print and electronic support documentation) to give the users options for statistically quantifying surface topography beyond rms roughness (including with fractal dimensions, for example). Surface skewness, or the coefficient of surface skewness, to be more exact, belongs to a family of mathematical functions that work with statistical distributions. There are others like it included in AFM image analysis software. It is up to the users to simply apply them and see what happens beyond rms roughness.

Fadrad Michael Serry
http://www.michaelserry.com/
serry@michaelserry.com

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¹STM is infrequently used in industrial applications, because it does not work with electrically non-conducting samples. This severely limits the kind of samples that are amenable to STM analysis.²The choice of symbols in the equations here is similar to those in the software package “Scanning Probe Image Processor“ from Image Metrology A/S, Lyngby, Denmark http://www.imagemet.com/.

³This is one definition of skewness; others exist. A list of some is available from Wolfram Research Corporation at http://mathworld.wolfram.com/skewness.html.

⁴Mandelbrot, Benoit B., and Hudson, Richard L., The (mis)behavior of markets: a fractal view of risk, ruin and reward, Profile Press, 2004, London, UK. ISBN 1861977654.

AFM is a powerful characterization tool for polymer science, capable of revealing surface structures with unprecedented spatial resolution. It is extremely useful for studying the local surface molecular composition and mechanical properties of a broad range of polymer materials, including block copolymers, bulk polymers, thin-film polymers, polymer composites, and polymer blends. 

In addition to remarkably high spatial resolution, another key advantage of AFM is simultaneous multichannel data acquisition. In acoustic AC mode, tip-sample force interactions cause changes in the amplitude, phase, and resonance frequency of the oscillating cantilever. The spatial variation of the change can be presented in height (topography) or interaction (amplitude or phase) images. A feedback system monitors the oscillating amplitude of the cantilever at each sample location and tries to maintain a set value (set-point) by moving the scanner up or down based on the surface morphology. 

While the vertical motions of the scanner are used to generate a topographic image, the actual oscillation amplitudes and the phase lag between the AC drive input and the cantilever oscillation output can also be collected simultaneously to produce the corresponding amplitude and phase image, respectively. It has been demonstrated by many research groups that phase contrast is very sensitive to differences in material properties, such as variation of mechanical and adhesive properties. 

The visualization of different components of heterogeneous polymer materials via AFM phase imaging has been demonstrated in numerous studies of block copolymers, semicrystalline polymers, and mesomorphic polymers. For instance, compositional mapping with AFM is often used for observations of microphase separation of block copolymers, which occurs at the sub-100 nm scale. In addition to the compositional imaging of multicomponent polymer samples, visualization of amorphous and crystalline components is an important application of phase imaging. Besides amorphous and crystalline forms, many liquid crystalline polymers such as poly(diethylsiloxane) (PDES) usually exist in a partially ordered or mesomorphic form, which can also be characterized by phase imaging. 

Polymer or plastic materials can be divided into two major groups, thermoplastic or thermosetting, based on their response to heat. Therefore, knowledge of polymer behavior at different temperatures is essential for many practical applications. Although quite a few macroscopic techniques, such as differential scanning calorimetry (DSC), dynamic mechanical analysis (DMA), and thermal mechanical analysis (TMA), are usually employed to probe the temperature changes of polymers, direct nanometer-scale visualization of polymers at different temperatures is still highly desired. With the development of both heating and cooling accessories, the use of AFM on polymer materials can be extended from ambient temperature to temperatures where polymer phase transitions occur. High-resolution AFM temperature studies can provide unique microscopic insight into polymer thermal behavior. 

We have documented many examples of the imaging of different polymer samples with the Agilent 5400 atomic force microscope that demonstrate its capabilities for visualizing important polymer nanostructures and monitoring structural changes caused by thermal transitions. The 5400’s outstanding thermal control is a rare feature for economically priced microscopes and the additional advantage of MAC Mode compatibility provides direct drive imaging in oscillatory mode in liquid and air. 

Please refer to the polymer-related application notes posted on our website. Detailed information about the 5400 atomic force microscope, thermal control, and MAC Mode can also be found on our website. 

Over the past two decades, the use of scanning probe microscopy to directly visualize electrochemical processes in situ at the molecular and atomic levels has increased dramatically. To demonstrate the high resolution and utility of ECSPM techniques for interfacial investigations, we have presented a number of original experiments (please refer to the application notes posted on our website).

In one instance, we conducted an in situ ECSTM experiment to watch the order-disorder transition of 2,2´-bipyridine (22BPY) on Au(111) surface under potential control. Individual bipyridine molecules in the ordered phase and their orientations were resolved, helping to understand the polymerization and ordering process of 22BPY at the molecular level.

2,2´-bipyridine was dissolved in 100 mM NaClO₄ to a final concentration of 1 mM. Deionized water (18.0 MW cm) was used throughout the experiment. A small Teflon cell used had an exposed electrode area of 0.28 cm². Ag/Ag⁺ was the quasi-reference electrode. Apiezon-wax-coated Pt/Ir tips had typical leaking current of 10 pA or less. Before each experiment, the fluid cell and electrodes were cleaned with H₂SO₄/H₂O₂ mixture and thoroughly rinsed with deionized water. After a gold substrate was hydrogen flame annealed, it was immediately transferred to a sample stage and covered with the electrolyte. Typical bias and setpoint current used were 200 mV and 0.2 nA, respectively.

The ordering process of 22BPY molecules on Au(111) at different potentials was then demonstrated. At 0.0 V (versus Ag/Ag⁺), molecules tended to be loosely in contact with the surface, randomly orientated. STM did not resolve either molecular rows or single molecules. At slightly higher potential, molecules started to bind to the surface and became observable.

When the surface potential was changed to 0.15 V, the adsorbate began to form short, parallel rows. At 0.20 V, over half of the molecules on the surface appeared to be ordered. Domains formed by groups of the same orientated molecular rows began to appear. At 0.27 V, the adsorbate showed long-range ordering. Three distinct orientations perfectly fit the underlying atomic lattice of the Au(111) surface. Domains and domain boundaries were visible. The measured chain-chain spacing was around 9 Å. Individual bipyridine molecules closely packed along polymeric chains were clearly resolved with a period of 3.3 ± 0.3 Å. The disorder-to-order transition was reversible and images were stable over several hours.

We then used in situ ECAFM to repeat the experiment of Cu underpotential deposition (UPD) on Au(111) with both molecular and atomic resolution.

The electrolyte was 100 mM H₂SO₄ containing 5 mM CuSO₄. A piece of Cu wire was used as the quasi-reference electrode. The AFM fluid cell had an exposed electrode area of 0.57 cm². Si₃N₄ cantilevers used had a typical spring constant of 0.5 nN/nm. The cell and electrodes were thoroughly cleaned before the experiment. Similar to the STM experiment, a gold substrate was flame annealed right before being covered with the electrolyte.

A cyclic voltammogram of Au(111) in 100 mM H₂SO₄ containing 5 mM CuSO₄ showed very distinct UPD peaks at 0.275 V versus Cu/Cu₂⁺ (DER = 9 mV). AFM images were acquired both below and above the peaks. At high potential (prior to Cu deposition), the atomic lattice of bare Au(111) surface was repeatedly observed. The unreconstructed Au atomic structure on the (111) plane with the familiar threefold symmetry was clearly resolved, showing an atomic spacing of 2.9 ± 0.2 Å.

Further ramping up the potential (to 0.70 V) did not significantly change the atomic image. Ramping down, however, showed a new lattice after passing the UPD peak. The measured lattice constant was 5.0 ± 0.3 Å and the orientation was 30º ± 1º relative to Au(111) lattice. These parameters suggested that the new lattice was (√3 × √3)R30º. When the potential was ramped below 0.060 V, the lattice disappeared and a full monolayer of deposited Cu was formed. When the potential was returned, the lattice reappeared. The (√3 × √3)R30º structure was very stable at a constant potential, indicating a strongly bound layer of molecules.

With continued advances in AFM instrumentation and in situ technologies, ECSPM is being utilized in an increasingly broad range of application areas.

We invite you to share any AFM-related comments, queries, suggestions, and ideas with us, as well as with your fellow researchers, on this blog.