In an earlier posting on this blog about surface roughness measurement with the AFM, I wrote that the rms roughness is too often the only numerical parameter that is computed and reported (February 4, 2008). I wrote about asymmetry in the height distribution, and how the rms roughness does not address it, and used skewness as an example of a parameter that does.

Here, I’d like to continue that thread, and explore another aspect of surface roughness the characterization of which requires computation beyond the rms roughness: the contribution of different spatial wavelengths to roughness.

The texture of a surface is characterized by the height of its peaks and the depths of its troughs. It is also characterized by the distances that separate the peaks and the troughs. This article is concerned primarily with those distances. Are there any dominant values of the spacing between peaks (or, alternatively, between troughs)? If so, what are they, and what is their relative order of dominance? Along which directions in the image are these dominant spacing to be found? So, we speak of dominant wavelengths (or dominant spatial frequencies) contributing to surface roughness, and we seek not only the value of these wavelengths, but also their directions in the surface texture.

Qualitatively, one surface may appear more wavy than another. One surface’s waviness may involve wavelengths shorter than those of another, making it look more rippled, even jagged. Qualitatively, roughness is nothing but waviness at very small wavelengths. One surface may look rougher than another if its dominant in-plane wavelengths are shorter.

All these and many more qualitative characteristics that we can visualize in an SPM image, we can also describe and compare quantitatively by using several features in SPM software. One of these features is called The Power Spectral Density (PSD) Analysis.

PSD analysis belongs to the frequency domain, and as such, it requires transforming the time domain or spatial domain data into the frequency domain, which is almost universally done using Fourier methods.¹

Any waveform in the time domain can be constructed from a linear superposition (an algebraic sum) of a collection of Sine and Cosine waves of different temporal frequencies or, equivalently, of different wavelengths. In an SPM image, the time translates into a distance (distance = time x scanning speed); it is therefore more meaningful to work with spatial wavelengths, rather than (temporal) frequencies.

The AFM image in Figure 1 shows two adjacent regions of a sample’s surface. We can use the AFM analysis software to zoom into each region and “planarize” it (remove slope, bow, …) in order to visualize the roughness better. This can be seen in Figure 2. There emerges a subtle but noticeable difference in the texture and topography between the two regions. This is a result of the way that these regions of the sample surface were processed, each differently from the other. We are interested in a way to possibly quantify the subtle differences between the two regions that we can visualize qualitatively. The region on the right appears to have a weak orderliness to it; there appears to be some faint correlation of the features across this region which is missing in the left region. To put it in a different way, the texture in the left region appears to have a more random character than the one on the right. In this particular software, SPIP Version 4.6.4.0, you can actually see this difference a little easier if you grab one of the small handle bars (arrows) that are attached to the color bars on the right of each image, and move it up and down the along the color bar repeatedly, but we are interested in quantifying the difference that we can visualize..²

Average X and average Y Power Spectral Density (PSD) plots of the two regions are shown in Figure 3 and 4. Although there are some differences in these plots (left region versus right region), there is nothing that immediately and clearly points to the presence of the order, however indistinct and dim, that we can see in the right region and that is absent from the left.

Figures 5 and 6 have what we are after. These figures show maps of the two dimensional PSD for each side of the image, and also a cross section, a profile of each map along a particular direction, the direction that is depicted by the rectangular zoom box in each two dimensional map. For the right side of the image, Figure 5 shows a strong peak at a wavelength of approximately 0.66μm. There are two things that are important about this data. First, this peak is completely absent in the profile along the same direction in Figure 6, which belongs to the left side of the image in Figure 1. It is absent in fact across the entire 180 degrees that you can sweep the profile zoom box around the center of the two dimensional map in Figure 6. This difference between the data in Figures 5 and 6 quantifies the visible, however subtle, difference in the image in Figure 1 between its right and left sides. Along the direction depicted, there is a relatively dominant periodicity, at wavelength of about 0.66μm, in the right side of the image, but not the left. It is visible upon close inspection that there appears to be some weak periodicity in the right side of the image, we see that and we were hoping to quantify it; we have. But the direction along which this periodicity resides is indeed unclear from close inspection. It was in fact surprising when I first saw it! This is the power of PSD analysis, in two dimensions. But there is more.

The second important thing about this data, the one that is not easy to show absent a sequence of profiles for which this blog article does not have the space, is that the peak in the profile in Figure 5 is present not only along the indicated direction, but also across a range of angles centered around that direction, and the wavelength itself varies by some 10% as you sweep across this range of angles. The peak is at its strongest at the angle shown in the profile. This spread of the angles and of the wavelengths is the way that the PSD analysis quantifies the uncertain and indistinct nature of the apparent order that we can visualize on the right side of the image: there is some kind of order, some sort of correlation, it is faint, it is subtle, but it is there, and the PSD analysis—in two dimensions—has quantifies it.

In an upcoming blog article, we will explore how this type of correlation can also be quantified using a different method, a method that belongs to the spatial domain and that is related to PSD in the frequency domain: the Autocorrelation method.

Figure 1. Two regions of a sample surface show subtle differences in their topography.

Figure 2. Each region in Figure 1 separately planarized by removing slope, curvature, and third order departure from planarity.

Figure 3. Top, average X and average Y Power Spectral Density (PSD) plots, and bottom, Isotropic Area PSD plot of the right region in Figure 1.

Figure 4. Top, average X and average Y Power Spectral Density (PSD) plots, and bottom, Isotropic Area PSD plot of the left region in Figure 1.

Figure 5. Two dimensional PSD map of the right region, and the average profile of it along a direction defined by the narrow rectangular white zoom box in the map. There is a dominant peak at wavelength 0.66μm.

Figure 6. . Two dimensional PSD map of the left region, and the average profile of it along the same direction as depicted in Figure 5. There is no peak (other than the one at position zero which carries no significant information.)

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

¹For more on Fourier methods applied to SPM images, please see an article that appeared on this blog on July 28, 2008.

²SPIP is Scanning Probe Image Processor, a product of Image Metrology Corporation of Denmark http://www.imagemet.com

Silver nanoparticles are known for their anti-microbial properties and have been used in bandages, socks (to prevent foot infections), and laundry detergent. Researchers are also looking into their use for food packaging and potentially even directly into foods. Iowa State University is researching this possibility — Click HERE to see the website.

Of course the flip side of nanoparticles is the safety concerns. Consumer Reports has weighed in on the subject — Click HERE to see the website.

Click HERE to see another Consumer Reports article specifically points out the risks of silver nanoparticles.

As a measurement expert and instrument supplier, we hope to do our part to unravel the mystery of what is dangerous and what is helpful. I’ll try to keep you updated on our efforts.

Click HERE if you want to read more about our particle analysis instruments.

Click HERE if you want to read an application note on manipulation of nanoparticles using an atomic force microscope.

Have you ever wondered what is going on electrically at the nanoscale?  Do you have a need to understand the impedance across a nanotube?  The good news - now you can find out.   Agilent engineers and scientists have now married together a network analyzer and an atomic force microscope, making it possible to make impedance and capacitance measurements at the tip of an AFM.  In fact, you can scan across a surface making these measurements. 

I remember visiting a number of research facilities shortly after Agilent announced its line of AFM’s and the big question from the researchers - “When will Agilent be able to make electrical measurements at the nanoscale?”  They were asking for worldclass measurements on objects that weren’t even visible through an optical microscope.  In effect - that day has arrived.   A whole new world of research is now available to us. 

To read more, visit the Agilent nanotechnology website and click on the image in the center of the page. 

Before it is modified and morphed into an SPM image, the underlying data, in a more primitive form, is one continuous, long sequence collected in the time-domain—a single one dimensional waveform that contains the variation of one dependent parameter, e.g. the AFM cantilever deflection, with the one independent parameter, time.

To create the image, this waveform’s independent parameter—its domain—is transformed from time into distance. Then the waveform is folded back and forth several times into a series of equal-length segments that are then successively aligned precisely adjacent to one another to form the image. As a result, the image is a new entity that has transformed the original one dimensional time-domain waveform into a two dimensional, spatial-domain waveform.

Every waveform in the time domain or spatial domain can be constructed from (or decomposed into) a superposition—an algebraic sum—of a series of simpler component waveforms that have different characteristics, such as frequency, amplitude, and relative phase. The component waveforms belong to a larger set, usually called the basis set, within which some members may be entirely absent from or are insignificant contributors to any given waveform. Sine waves are most commonly used as a basis set, but this is not the only choice available.¹

The Frequency Domain
Frequency domain analysis involves transforming the time or spatial domain data—the waveform—into the frequency domain, then measuring and comparing the amplitudes and phases of the component waves at the frequencies that make up the waveform. When sine waves constitute the basis set, this is called Fourier analysis, and involves some implementation of the Fourier transform.

In scanning probe microscopy (SPM) imaging, frequency domain analysis can be performed on time-domain data as raster scanning proceeds, or on recorded data (images or spectroscopy plots). Three classes of instruments are often used to transform time-domain data into the frequency domain: spectrum analyzers, network analyzers, and dynamic signal analyzers. Each instrument class has its own advantages over the others for certain types of measurements. Network analyzers, for example, allow phase measurements, while spectrum analyzers generally do not. On the other hand, spectrum analyzers can measure smaller signals because they usually have lower noise floors than network analyzers. Dynamic signal analyzers allow modal analysis, while the other two classes of instruments do not. Nonetheless, all three classes are frequently interfaced with an SPM to analyze its time-domain data.

In an SPM image, time translates into space via the scan speed: time series of data points are recorded versus spatial sampling points. Usually, the fast scan direction is designated “X”, the slow “Y”. Frequency domain analysis of SPM images is carried out in the SPM software, and concerns spatial wavelengths, rather than frequencies. Hereafter, “frequency” will continued to be used per standard signal processing terminology, but it must be understood that in image processing, including SPM images, (spatial) wavelength, not frequency, is the natural parameter with which to work.

Why the Frequency Domain?
Why would we want to study an SPM image in the frequency (wavelength) domain?

Perhaps the simplest, most intuitive hint to the answer is this: Periodicity is inherent in nature, and influences the way it works. The properties of a sample surface often depend on the periodicities (wavelengths) characteristic to that surface.

In an AFM image, we may consider a line in any direction, as shown in Figure 1, and seek the variations of the topography along this line. We may be interested not only in the rms roughness, but also in the spatial frequency characteristics of the roughness.

Figure 1. Spatial-domain visualization and analysis. AFM image of a reference sample, and the variation of the topography along a line of arbitrarily chosen direction.

We may also want to consider all directions simultaneously, and inquire about the variations of topography across either the entire image or a subsection of the image. But again, we may want to characterize the area of interest beyond measuring parameters such as rms roughness, skewness, and maximum and minimum height, all of which are wavelength-independent and direction-neutral. We may want to find, for example, any preferred direction(s), or quantify any spatial in-plane symmetry that the image holds.

Frequency domain techniques provide numerous ways to extract information from an SPM image. While periodicity at some frequencies may be readily visible in the spatial domain (in the image,) at other frequencies it may be faint and difficult to decipher. The image in Figure 1, in which the periodicity is 10mm along two orthogonal directions, is a case in point. When we transform the spatial domain data along the virtual cross-section line (indicated by the arrow) into the frequency domain via Fast Fourier Transform (FFT), the sine waves that contribute most to the spatial domain data stand out at frequencies that correspond to wavelengths 10μm, 3.33μm, 2μm (Figure 2).

Figure 2. Frequency domain analysis. Fourier spectrum of the cross-section data in Figure 1.

The latter two are called the third and the fifth harmonics of the fundamental wavelength at 10μm. Higher order odd-numbered harmonics (beyond the fifth) also contribute relatively strongly compared to their nearest even-numbered harmonic neighbors, but not as strongly as the third and fifth harmonics. This kind of information is valuable in understanding the structure of the surface, in relating that structure to the properties and functionality of the surface, and also in filtering, as described next.

Filtering
Frequency domain techniques enable the selective admittance of some frequencies to the waveform and the exclusion of others. This is called frequency domain filtering, the results of which can be quite impressive. Figure 3 shows what happens to the AFM image and the cross-section data from Figure 1 when we filter it. We use FFT to transform the image into the frequency domain, filter out the high-frequency components (the higher harmonics) while keeping the low-frequency ones (in this case, only the fundamental frequency), and then use inverse Fast Fourier Transform to reconstruct the image from the low-pass filtered spectrum.

Figure 3. Low-pass filtered image and cross- section, same location as in Figure 1.

On the other hand, if we filter out the low frequencies while keeping the higher ones in the spectrum, and then inverse transform to reconstruct the image, we arrive at the results depicted in Figure 4. The edges of the square pits in the original image are visible, but the height variation across the image has largely disappeared (notice the height scale in the cross-section data, and compare with that in Figures 1 and 3).

Figure 4. High-pass filtered image and cross- section, same location as in Figure 1.

SPMs are extremely susceptible to noise from all types of sources. One of the most frequent uses of analyzing SPM images in the frequency domain is identifying spatial wavelengths (and therefore the temporal frequencies) of the noise source(s), at a single frequency or a collection of frequencies. Once the frequencies are known, the noise source(s) may be easier to identify and suppress, leading to enhanced quality of the subsequent SPM images and accuracy of the measurements made on the same.

More Power in Frequency Domain Techniques
Some of the most powerful methods that arise from understanding and working with the frequency domain involve mapping a relatively difficult operation to be performed on spatial (time) domain data onto a relatively easy operation to be performed on the corresponding frequency domain data. One example is how the relatively complex convolution operation in the spatial (time) domain is substituted for by the relatively simple multiplication operation in the frequency domain, a subject that we shall visit in the near future in this blog.

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

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¹The choice of the working basis set is made in part by considering the symmetries and the boundary conditions that apply to the waveform in the time domain or the spatial domain. See for example Chapter 1, in “Spatial Vision” by R L. De Valois and K. K. De Valois, Oxford Psychology Series, 14, Oxford Science Publications, Oxford University Press, US 1988.

Remember the game hide and seek?  If you played this game outside, as a “hider” it was important to find a really big tree so the “seeker” had no chance of spotting you.  As the seeker, you always wondered if anyone was hiding behind the big trees. 

The same thing happens in the world of particle analysis.   The particle analyzers can very easily spot the big particles and measure their size.  However, you always wonder if a small particle is hiding behind the large one? 

Not any more!  You might want to check out this application note.  The advantage in hide and seek now goes to the seeker with a new technological breakthrough in particle analysis.  I’m not going to shamelessly hype any products but you should probably at least take a look at this new technology. 

Application Note is found at:  http://www.chem.agilent.com/temp/radF9367/00001916.PDF

Product information at: http://www.chem.agilent.com/Scripts/PDS.asp?lPage=62634

I remember as a kid we were sometimes allowed to mix up a glass of chocolate milk after school.  If we used too much of the chocolate powder or we didn’t mix it properly, we ended up with a bunch of black goo in the bottom of the glass.   And yet when we had the little containers of chocolate milk at school, it didn’t have that problem.  Why was that?

Chocolate milk reminds me a little of particle analysis and the zeta potential of colloids.  Agilent has recently introduced its Series 7000 line of particle analyzers.  (See www.agilent.com/chem/particles for more info) One of the analyzers - the Agilent ZetaProbe - is all about ensuring the stability of the suspension.  As I talk with customers working with colloids not everyone understands the power of using zeta potential to ensure the stability of the suspension. 

If you’re someone dealing with suspensions, let me recommend a couple of resources to learn more about zeta potential.  

Applications notes explaining zeta potential: http://www.chem.agilent.com/scripts/LiteratureResults.asp?iprodinfotype=4&imodel=1765 

e-seminar presentation given by Dr Richard O’Brien, a pioneer in the field of zeta potential: http://www.chem.agilent.com/scripts/LiteratureResults.asp?iprodinfotype=4&imodel=1765

My recommendation - mix yourself a glass of chocolate milk - and read up on zeta potential.  Let me know how it goes! 

Although I’m not a medical doctor, I’m very intrigued with the possibility of nanoparticles becoming the drug delivery mechanism for the future.  Delivering small doses of medicines directly to cancer cells would eliminate the painful chemotherapy that many now have to suffer through.  I’ve had several friends who have gone through this misery and it’s something that I hope researchers and doctors can figure out.

The key is really to use the right sized particle, coated with a molecule that will attract itself to the diseased cell.  Once inside the cell the nanoparticle can deliver enough medicine to kill just that cell and not the healthy cells that may be around it. 

Agilent announced at Pittcon a particle size spectrophotometer that can measure the particle size distribution in the 5 nm to 15 um range - the sweet spot for these drug delivery nanoparticles.   Using our proven UV-VIS spectrophotometer technology the instrument is able to measure and display a wide distribution of particles in suspension in only 5 seconds.   Getting the right-sized nanoparticles should help researchers find the drug delivery mechanism of the future.

If you are looking for a good article on this subject check out this website — http://www.expresspharmaonline.com/20080415/healthcare01.shtml

If you want to learn more about the Agilent 7010 Particle Size Spectrophotometer, check out this website — www.agilent.com/chem/particles

Last week I attended the Pittcon show in New Orleans.  It’s the largest chemical analysis show in the country with all the major analytical equipment suppliers in attendance.   New Orleans went all out to host this large event and it was good to see the city coming alive after Hurrican Katrina. 

From a nanotechnology perspective I was involved in the launch of Agilent’s new Materials Science Solution Unit (MSSU).   The purpose of this group is to focus on the transition of nanotechnology from pure research to new materials made up of nanoscale structures.   Our CEO, Bill Sullivan, hosted a press conference where he introduced Mike Gasparian, the VP / General Manager, of the MSSU.  Mike talked about the newest additions to the Agilent nanotechnology family of products - particle analyzers and optical fluorescence microscopes - and the acquisition of Colloidal Dynamics (known for their electroacoustic zeta potential measurements) and TLL Photonics (known for a revolutionary new digital microscope). 

I’ll have further blogs on these topics in the future - stay tuned.

If you want to know more you should click on TLL Photonics link or visit the Particle Analysis website at www.agilent.com/chem/particles

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.

A colleague and I were discussing some ideas for this nanotechnology blog when we jokingly commented that if we wrote it about the Super Bowl we would probably get a lot more people tuning in to read it.  Out of sheer curiousity I ran a quick Google check on football and nanotechnology and found this interesting article about a contest being run by the American Physical Society.  http://blogs.zdnet.com/emergingtech/?p=807

One can win the smallest trophy ever made and a $1000 in cash by creating a video that demonstrates some aspect of physics in American football.  The winner will be announced on Super Bowl Sunday.  Sounds like fun!  Time to dig out your camcorder and go to work. 

The trophy will be built at the Cornell NanoScale Facility (CNF) on a silicon wafer in the shape of a football field.  In case you win the world’s smallest trophy - Agilent would be very happy to sell you an atomic force microscope so you can see it!  (How’s that for a shameless plug!)