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It is well known that determination of concentrations of materials in different solutions is an important step for investigation of the under-studied solution. Usually, photometric techniques are used due to this fact that they are accessible and cost-effective options.

Generally, absorption of irradiated light to a solution by the presence molecules is the base of (spectro)-photometric techniques. In UV/Vis spectroscopy visible and ultraviolet light uses for detection of concentration of a solution.

Spectroscopy is a science that studies the interaction of electromagnetic radiation with matter. In such interactions, electromagnetic radiation can be thought of as a set of separate energy packets called photons. The dual property of electromagnetic radiation as a particle and a wave is not only non-existent but also complementary. According to the theory, electromagnetic radiation is made up of two components, electric fields and magnetic field. These fields are propagating the wave in the environment, on the environment and also perpendicular to the wave propagation (Figure 1).

The electric field of electromagnetic radiation causes phenomena such as transmission, reflection, refraction, and absorption when it interacts with matter. The magnetic field of electromagnetic radiation is also effective in the process of absorbing waves related to radio frequencies in nuclear magnetic resonance. Therefore, here only the electric field of electromagnetic radiation is examined due to its effectiveness in the above phenomena.

As was mentioned previously, determination of concentrations of materials in different solutions is an important step for investigation of the under-studied solution. Usually, photometric techniques are used due to this fact that they are accessible and cost-effective options.

For the calculation of an analytic concentration, the Lambert-Beer law form the basis can be used as follow:

1. Transmission or transmittance (T) = I/I₀
   
2. Absorbance (A) = log (I₀/I)
   
3. Absorbance (A) = C x L x Ɛ => Concentration (C) = A/(L x Ɛ)
   

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Fourier transform infrared spectroscopy (FTIR) is an experimental test to identify organic/inorganic materials. But for understanding and interpretation of FTIR results, the principles of the analysis should be studied.

Principles of FTIR

To better interpretation of FTIR results, it’s useful to know about the principles of the test.

Generally, by introducing infrared radiation (IR) to a material, FTIR’s detectors detect a sample’s absorbance of infrared light at various wavenumbers to determine the material’s chemical structure. FTIR spectrometer converts the experimental data from the broadband source to the absorbance level at each wavenumber.

This method can be employed for solids, liquids, and gaseous samples. Some times, the weight of samples required for a viable analysis is very low and most analyses can be conducted with little sampling process.

How to interpret FTIR spectra

The X-Axis

The horizontal axis shows the IR spectrum, which represents the intensity of the IR spectra. The absorbance peaks, can be attributed to the various vibrations of the sample’s bonds exposed to the IR energy of the electromagnetic spectrum. Usually, the wavenumber on the IR spectrum is drew between 4000 to 400 cm-1.

The Y-Axis

The vertical axis shows the amount of IR absorbance/transmittance by the under-studied material.

The Absorbance Bands

Generally, absorbance bands can be derived into two groups: Group frequencies and fingerprint frequencies.

Group frequencies are assigned to small groups of bonds such as C-H, O-H, and C=O. These types of bonds are usually observable at 1500-400 cm-1 in the IR spectrum and they are typically unique to a specific bond or functional group in a structure.

For fingerprint frequencies, these are assigned to the chemical structure as a whole. These types of absorbances are usually observable at 400-1500cm-1 in the IR spectrum. Therefore, this region is less reliable for identification, however the absence of a peak is often more indicative than the presence of a peak in this region.

How to analyze FT-IR Spectra

Generally, interpretation of FT-IR spectra starts at the high frequencies end to identify the presence functional groups. The fingerprint regions are then investigated to identify the chemical bonds.

Still Curious About FTIR Analysis?

FTIR is a useful technique for manufacturers and researchers of various industries. If you have more questions about the analysis or are wondering if it may be a fit for your testing needs, contact us for a quote.

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Our XRD interpretation includes: 
1. Phase determination 
2. Determination of diffracted planes
 3- Calculation of crystalline size and microstrain
 4- Whatever your request
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XRD is a non-destructive test method used to analyze the structure of crystalline materials. XRD analysis, by way of the study of the crystal structure, is used to identify the crystalline phases present in a material and thereby reveal The chemical composition information. JCPDS does not exist now. It has not existed since 1978. It is now known as ICDD. These particular files have never been, are not, and never will be free; it a commercial only database. There are other free databases, however. Vikas has given you a starting point.

Powder Diffraction File is a trademark of the “JCPDS (Joint Committee on Powder Diffraction Standards)-International Centre for Diffraction Data”.In 1978, the name of the organization was changed to the “International Centre for Diffraction Data” in order to highlight the global commitment of this scientific endeavor. Here, you need to purchase the database.

Some free databases are collected:

1. COD (Crystallography Open Database):

COD is an open-access database, and you can freely obtain all data contained in it. You can download cif files and then you can use mercury to plot structure models and save reflection list and xrd calculated pattern.http://www.crystallography.net/cod/search.html

2. The American Mineralogist Crystal Structure Database:

This site is an interface to a crystal structure database that includes every structure published in the American Mineralogist, The Canadian Mineralogist, European Journal of Mineralogy and Physics and Chemistry of Minerals, as well as selected datasets from other journals. The database is maintained under the care of the Mineralogical Society of America and the Mineralogical Association of Canada, and financed by the National Science Foundation.http://rruff.geo.arizona.edu/AMS/amcsd.php

3. DASH: (Cambridge Structural Database System (CSDS)):DASH is a versatile and interactive package for solving crystal structures from powder diffraction data. DASH solves structures by simulated annealing of structural models to indexed diffraction data and features a helpful wizard to guide you through the entire structure solution process.https://www.ccdc.cam.ac.uk/solutions/csd-materials/components/dash/

Some of Paid databases:

1. The International Centre for Diffraction Data® (ICDD®):

ICDD (JCPDS is now called ICDD) is a non-profit scientific organization dedicated to collecting, editing, publishing, and distributing powder diffraction data for the identification of materials. The membership of the ICDD consists of worldwide representation from academe, government, and industry. The Powder Diffraction File™ (PDF®) is the only crystallographic database that is specifically designed for material identification and characterization. It is an analysis system that is comprised of crystallographic and diffraction data. The only crystallographic database organization in the world with its Quality Management System ISO 9001:2015 certified by DEKRA.http://www.icdd.com/

2. HighScore Plus:The ideal tool for crystallographic analysis and more. Whether you are interested in improved process control, or doing research and development, understanding your materials starts very often with understanding the powder diffraction pattern. After identification of all phases present in your sample with Malvern Panalytical’s HighScore, this all-in-one software suite with the Plus option continues to support you with your analysis. Whether your focus is on quantification with or without the Rietveld method, profile fitting, or pattern treatment; HighScore Plus is the solution and helps you performing your daily analyses.https://www.malvernpanalytical.com/en/products/category/software/x-ray-diffraction-software/highscore-with-plus-option

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1. NMRshiftdb

NMRshiftdb2 is a NMR database (web database) for organic structures and their nuclear magnetic resonance (nmr) spectra. It allows for spectrum prediction (¹³C, ¹H and other nuclei) as well as for searching spectra, structures and other properties. The nmrshiftdb2 software is open source, the data is published under an open content license. The core of nmrshitdb2 are fully assigned spectra with raw data and peak lists (we have pure peak lists as well). Those datasets are peer reviewed by a board of reviewers. The project is supported by a scientific advisory board.

nmrshiftdb2 is part of the NFDI4Chem initiative and will provide a component for a curated repository there. Please consult the documentation for more detailed information.

See: https://nmrshiftdb.nmr.uni-koeln.de/portal

2. ACD/NMR

ACD/NMR Workbook Suite is a comprehensive NMR software application with an intuitive interface. It features a full suite of advanced processing, analysis, and databasing functionalities for 1D and 2D NMR data from all major vendor formats. NMR Workbook Suite is built upon cutting-edge algorithms for the most reliable NMR data interpretation. It is designed to streamline routine NMR workflows, simplify structure characterization, and much more.

Powerful NMR Interpretation Software | Highlights

• Import and process 1D and 2D NMR data from all major instrument vendor formats in a single collaborative platform
• Process NMR data manually or automate routine processing workflows—Fourier transformation, calibration, peak picking, integration, multiplet analysis, etc.
• Synchronize peak picking and assignments across datasets within a project
• Confidently verify structures with 3 different verification levels
• Perform targeted analysis of known mixture components and optimize untargeted mixture analysis workflow
• Perform Conformational Analysis using NOESY/ROESY spectra
• Create comprehensive multiplet reports and publication-ready data
• Store, manage, and share live NMR spectra

Synchronize peak picking and assignments across NMR datasets using NMRSync—our game-changing technology. Plus, the associated peaks from NMRSync, NMR prediction, and connectivity-based algorithms are automatically used to only identify the assignments that match all data. This quick and accurate peak picking and assignment workflow helps you to maximize your productivity in the following ways:

• Use any peak in any spectrum to initiate NMRSync
• Integrate a peak in any spectrum and all related peaks in the 1D and 2D NMR spectra of that dataset will be identified and linked in real time
• Automatically resolve overlapping ¹H and ¹³C peaks from 2D NMR data
• Receive immediate color-coded feedback on the best assignment for instant decision-making purposes

NMR Workbook Suite includes three levels of structure verification that evaluate alternative structures to varying degrees for added flexibility in your NMR analysis. This ensures the best structure that matches the experimental NMR data is confirmed with much less time and effort than manual interpretation.

• Determine how well your proposed structure matches the datasets in your NMR project with single structure verification
• Generate a specified number of alternative structures, based on the user-defined proposed structure, and evaluate whether they are a better match to the NMR dataset using Combined and Concurrent Verification
• Generate and view every alternative structural and cis/trans isomer that matches the experimental data in real-time using Unbiased Verification for an absolute level of confidence. This workflow eliminates the user bias and ensures the assigned structure is indeed the best structure that fits the experimental data.

See: https://www.acdlabs.com/products/spectrus/workbooks/nmr/

3. See: http://www.cheminfo.org/Spectra/NMR/Predictions/1H_Prediction/index.html

4. See: https://www.nmrprocflow.org/

5. See: https://chem.washington.edu/facilities/data-processing

6. See: https://www.cgl.ucsf.edu/home/sparky/

7. See: http://www.nmrdb.org/about/

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This handout relates the basic theory of NMR described on the theory web handout with spectra of real molecules and how to deduce structure from the spectra. Before reading this handout, you need to be thoroughly familiar with all of theory concepts that were described.

1.0 The NMR spectrum.

1.1 Because different amounts of electron density are around different non-eqivalent nuclei, the different non-equivalent nuclei in a molecule are experiencing slightly different net magnetic fields in an NMR experiment (Review Section 5.2A of the theory handout). Recall also that the difference in energy between the two allowed spin states (+1/2 and -1/2 spin states) of a spin 1/2 nucleus (like in 1H and 13C nuclei) depends on the exact magnetic field felt by the nucleus (Review Section 2.3Cin the theory handout). Recall further that in the NMR experiment, when and only when nuclei are irradiated with electromagnetic radiation of energy that exactlycorresponds to the energy difference between the +1/2 and -1/2 spin states, the nuclei absorb the energy and the NMR spectrometer measures this absorbance (Review section 3.1 of the theory handout). The absorbance of energy to convert a nucleus from a +1/2 to a -1/2 spin state is referred to as “resonance” of that nucleus.1.1A The key conclusion is that nuclei with different electron densities have +1/2 and -1/2 spin states that differ in energy by differing amounts, so these nuclei will absorb electromagnetic radiation of different frequencies in the NMR experiment.

1.1B Nuclei surrounded by greater amounts of electron density will be more shielded from the external magnetic field, so they will absorb electromagnetic radiation of lower energy, that is, lower frequency (energy is proportional to frequency). You may want to review Section 5.2A of the theory handout again.

1.1C The converse is also true, namely that nuclei surrounded by lesser amounts of electron density will be less shielded (referred to as being “deshielded”) from the external magnetic field, so they will absorb electromagnetic radiation of higher energy, that is, higher frequency(energy is proportional to frequency).

1.1D The three most important factors influencing the electron density around a hydrogen nucleus are: (i) adjacent electronegative atoms remove electron density; (ii) hybridization of the attached carbon atom, increasing shielding is observed in the order sp2, sp, sp3; (iii) adjacent pi bonds are deshielding, which relates to (ii).1.2 An NMR spectrum is a plot of absorbance versus frequency.

1.2A To make different spectra directly comparable, a standard is used for all NMR spectra. For 1H NMR spectra, the standard is called tetramethylsilane (TMS) and a small amount of TMS is usually added to any 1H NMR sample.

1.2B Magnets of different strengths lead to absorbance of electromagnetic radiation at different frequencies for the same nucleus, meaning that if simple frequency were plotted in an NMR spectra, you could not compare spectra taken of the same sample on machines with different magnet strengths. To solve this problem, the frequency of absorption plotted on NMR spectra are corrected for the magnet strength. In addition, frequency is correlated to the reference compound TMS. The frequency at which TMS absorbs is defined as 0 frequency by convention. In the NMR spectrum, absorbance frequencies of electromagnetic radiation are plotted as chemical shift (d) listed in units called parts per million (ppm) that is defined by the following equation:

1.3 The bottom line to this entire section is that the hydrogen atoms of different functional groups (methyl groups, -CH2- groups, aldehyde -C(O)H, alkene C-H, etc.) have characteristic chemical shifts, i.e. absorbance frequencies. These characteristic chemical shifts are collected in tables such as Fgure 13.8 and Appendix 4 of your book. From the chemical shift information, you thus know what functional groups are present in a molecule.

1.4 Chemically equivalent hydrogen atoms will have the same chemical shift and therefore give rise to the same signal. This is why we defined equivalent atoms in Section6.1 of the theory handout. Non-equivalent groups of hydrogens will have different chemical shifts. Thus, you will have as many different signals in an NMR spectrum as there are chemically non-equivalent groups of hydrogen atoms.

2.0 The nuclear spin of hydrogen atoms creates a magnetic field that influences the chemical shift of nearby hydrogen atoms (Review Sections 5.1 and 5.2).

2.1 Nuclear spin magnetic fields will influence hydrogen atoms that are three or fewer bonds away from each other in the same molecule. Hydrogen atoms that are four or greater bonds away usually do not influence each other.

2.2 A hydrogen atom with a nuclecus in a spin state of +1/2 produces a slightly different magnetic field than a one in a –1/2 spin state.

2.3 Even in a strong magnetic field, across a population of molecules, there is only a very slight excess of nuclei in the +1/2 spin state.

2.4 Putting all of these ideas together means the following: Consider a hydrogen X adjacent (three bonds away) to another hydrogen Y in a molecule. In around half of the molecules in the NMR sample, hydrogen X feels the magnetic field from a Y with nuclear spin of +1/2. The other half feel from Y a nuclear spin of –1/2. Thus, when you look at the spectrum, there are actually two different, but closely spaced peaks as the signal for hydrogen X. This phenomenon is called “spin-spin” splitting, and the distance between the two signals for X is called the “coupling constant”, often denoted as “J”. Similarly, the signal for Y actually has two peaks because of spin-spin splitting by X.

2.5 Consider a –CH2- group adjacent to a hydrogen X. Both of the hydrogen atoms in the –CH2- are chemically equivalent and could be either in the +1/2 or –1/2 nuclear spin state. Thus, there are three situations possible: i) +1/2,+1/2; ii) +1/2,-1/2, which is the same as –1/2, +1/2 and iii) –1/2,-1/2. Thus, there are actually three different magnetic fields that are felt by X in molecules of the sample, in a 1:2:1 ratio. Thus, the signal for hydrogen X is split into three peaks in a 1:2:1 ratio.

2.6 The same holds for a –CH3 group, that will split an adjacent hydrogen signal into four peaks, with a 1:3:3:1 ratio. You should verify this for yourself by making all the possible combinations of nuclear spins for the three equivalent hydrogen atoms of a methyl group.

2.7 In the general case, N equivalent hydrogen atoms will split an adjacent signal into (N+1) peaks, with relative ratios that are predicted by Pascal’s triangle (Figure 13.16 in the book).

3.0 Following the same logic, the splitting should multiply if a single hydrogen atom is adjacent to hydrogen atoms on either side. Think about combining all the possible nuclear spin states for these nearby sets of hydrogen atoms. Thus, if you have a hydrogen atom X between one –CH2- and one –CH3 group, it should be split into an amazing (2+1) x (3 + 1) = 12 signals because there are that many different combinations of +1/2 and -1/2 spins possible.

3.1 Thus, if the coupling constants (J) from the –CH2- and –CH3 groups are significantly different from each other, then 12 peaks will be observed as the signal for hydrogen X.

3.2 However, in practice, coupling constants (J) are pretty close to the same value for almost all sets of hydrogen atoms in organic molecules, simplifying the splitting pattern, since now many of the twelve peaks will overlap with each other. What this means is that for almost all the spectra you will see, if a hydrogen X is surrounded by N hydrogen atoms, the signal for X will be split into only (N+1) peaks, no matter how those N hydrogen atoms are grouped in terms of sets of equivalent atoms. Thus, what is actually seen for the example above is that the signal for X would appear in the spectrum to be split into 2 + 3 + 1 = 6 peaks, not 12, peaks. This is the so-called “N+1” rule.

3.3 The diagram below shows these two different situations. When nuclei from hydrogen atoms Z and Y split the signal for hydrogen X with very different coupling constants (notice how the coupling constant J for the red Z hydrogen nuclei is larger than J for the blue Y hydrogen nuclei), all twelve peaks are spread out and identifiable. Below that is shown the situation in which the coupling constants are the same for nuclei of both Z and Y, so only 6 peaks are actually observed in the signal for hydrogen X due to extensive overlap. This latter case, with six peaks, is what you will almost always see in reality since coupling constants tend to be similar in organic molecules.

3.3 The above explanation of splitting can confuse students for a while. The important point is that in the example given, you see 6 different peaks in the spectrum (N+1 rule) even though there are really 12 peaks produced, it is just that several of them are on top of each other because the coupling constants are the same. For alkyl groups in organic molecules, the coupling constants are generally the same so you will almost always see the fewer peaks, corresponding to the simple N+1 rule, rather than the greater number of peaks derived from the multiplication rule.

3.4 The bottom line here is that by seeing how a given signal is split, you can figure out how many hydrogen atoms are adjacent on the molecule, namely the number of peaks in the signal minus 1. From this information you can piece together what a molecule looks like if you know how many atoms of each type are present (i.e. the molecular formula such as C4H10N2O). You get the molecular formula information from something called a mass spectrum, described later in the text. Molecular formulas will be provided to you in homework or test questions.

4.0 For a given signal, integrating the signal (include all splitting peaks for a given signal) gives you a relative value that is proportional to the number of equivalent hydrogen atoms that gave rise to the signal. Thus, by looking at the integration values, you can deduce how many of each type of equivalent hydrogen atoms are in the molecule. For example, a -CH3 group would have a signal that integrates to a relative value of 3 (no matter how the signal is split), and a -CH2- group would have a relative integration of 2, etc. Note that sometimes integrations are simply given as absolute numbers, and you must find the common factor to deduce how many hydrogen atoms are represented by each integration value.

5.0 Putting it all together: How to deduce a structure from an NMR spectrum. First, you must be given the molecular formula, so you know how many of each type of atom are present. Second, count the number of different signals and their relative integrations to see how many different sets of equivalent hydrogen atoms are in a molecule, and how many of each set are present. Compare the chemical shifts of each signal to tables to identify what functional groups are present. Finally, use the signal splittings to determine which hydrogen atoms must be no more than 3 bonds away from each other.

6.0 For alkenes, the pi bond prevents bond rotation so the different hydrogen atoms on an unsymmetrical alkene are not equivalent, so they all have different signals, and splitting follows the multiplicative rule (the coupling constants are usually significantly different for geminal vs. cis. vs. trans relationships).

7.0 For hydrogens in a -CH2- group adjacent to a chiral center, the two different H atoms are no longer equivalent, because even with bond rotation, the two hydrogens are never in the same environment with respect to the groups on the adjacent chiral center. Thus, each H of -CH2- group adjacent to a chiral center usually has its own signal in the NMR spectrum.

8.0 There is a great deal more to NMR than this, I am only trying to give you the basics here.

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1. XPST

XPST is a program package for the analysis of X-ray Photoelectron Spectroscopy (XPS) data. It includes various graphical interfaces as well as commandline functions to facilitate the workup of XPS data.

When a XPST fit project is started, a corresponding subfolder with all required data is generated and saved within the Igor experiment. You can generate fit templates and you can export entire fit projects to share them with your co-workers. As a special feature, a flexible multiplet function was implemented to facilitate a convenient analysis of complex spectra. XPST can handle any number of peaks.

There is also a youtube channel with tutorials.

Several changes in the newest version of XPST were made according to this book about programming Igor.

Installation

XPST was initially developed with Igor 5, but a major revision was made with Igor 7. XPST works also nicely with Igor 8. The Igor 7/8 version is not compatible with Igor 6. If you still run Igor 6, you have to download a previous release.

• Unpack the .zip file
• Copy/Move the folder ‘XPST’ to the folder ‘Igor Procedures’ in Igor’s main folder
• Copy/Move the folder ‘XPSTHelp’ to the folder ‘Igor Help Files’ in Igor’s main folder
• Restart Igor

Before you upgrade to a newer version of XPST, please remove all files associated with your old version from Igor’s folders.
 

Stay in touch …

If you want to stay informed about updates or other issues, just send a blank mail to: xpst_update@freenet.de
This is not a support contact – it only serves to keep you informed about changes.
You can sign off anytime you want using the same address.
 

More functions …

Besides the graphical interfaces, XPST comes with several commandline functions. For example:

• WhereIs() …. returns the absolute path of a selected wave (so it can be easily found)
• WaveOverlap() …. computes the overlap of two selected waves and saves it to a new wave
• CursorCut() …. cuts out regions from a selected wave

Limitations

• Data with ‘kinetic energy’ as x-axis (in waveform format or not) can not be analyzed with the Fit Assistant. Only a positive ‘binding energy’ scale works – however, this should be the most common case. If there is a strong demand for kinetic energies, it could be implemented in future versions.

See: https://www.wavemetrics.com/project/XPStools

2. CasaXPS

CasaXPS processing software offers powerful analysis techniques for both spectral and imaging data. The system originally designed for XPS and Auger data now offers features covering a wide range of analytical techniques including ToF SIMS, dynamic SIMS and many more.

Features in CasaXPS include:

·         Full quantification including transmission correction.

·         Configurable quantification reports.

·         Background type ranging from the standard Linear, Shirley and Tougaard to user-defined cubic-spline approximations.

·         Asymmetric and symmetric line-shapes: Doniach-Sunjic, Voigt, Gaussian-Lorentzian and line-shapes defined from data.

·         Easy-to-use propagation of processing, annotation and peak models to other data.

·         Batch processing for repetitive tasks, including configurable processing, display layout with automatic printing and report generation.

·         State-of-the-art image processing for XPS spectromicroscopy offering quantified chemical-state XPS images.

See: http://www.casaxps.com/

3. Originlab

Origin provides powerful and versatile tools such as Peak Analyzer, Quick Peaks Gadget, Integration Gadget, etc. for baseline correction, peak detection, peak integration and peak fitting.

Origin provides baseline detection and subtraction. Key features include:

• Create baseline with many baseline modes
  • Commonly used baseline types such as Constant, Straight line, Use Existing Dataset, End Points Weighted, Min&Max
  • Automatically find anchor points for baseline, modify the points, create baseline by interpolation or fitting
  • Create baseline for XPS using Shirley or Tougaard method and adjust corresponding parameters to optimize
  • Create baseline using Asymmetric Least Squares (ALS) Smoothing method and adjust corresponding parameters to optimize
• Subtract baseline

Origin allows you to search for peaks including hidden (“convoluted”) peaks and filter out unwanted peaks or manually add or remove peaks.

• Savitzky-Golay smoothing on the spectrum before peak finding

In Origin, you can integrate data with multiple peaks, to obtain peak areas, FWHM and other peak characteristics. Baseline subtraction is supported before peak integration. 

Available options include: 

• Directly select desired data range on graph
• Instantly view results of peak area and FWHM
• Subtract baseline from peak data
• Auto detect peak positions
• Auto determine peak widths for overlapped peaks
• Individually set peak widths
• Fit peaks and obtain fitted peak areas

Origin provides many tools to perform peak fitting: 

• Quick Peaks Gadget  : Visually correct baseline, find and fit peaks
• Multiple Peak Fit: Manually pick peak positions and fit peaks with same function. No baseline correction
• Peak Analyzer: Correct baseline, find peaks and fit by Peak Analyzer wizard
• Nonlinear Curve Fit Dialog: Fit multiple peaks with replicas in the nonlinear curve fit dialog

Available options for peak fitting include:

• Fit peaks with built-in or user-defined functions
• Fit multiple peaks with different functions
• Control fitting process using bounds and constraints
• Fix or share peak parameters
• Vary baseline parameters along with peak fitting

See: https://www.originlab.com/index.aspx?go=Products/Origin/DataAnalysis/PeakAnalysis#Peak_Fitting_PRO

4. XPSPEAK

Free, fully featured, software for the analysis of XPS spectra written by Raymund Kwok.XPSPeak is a XPS Peak Fitting Program.The portable app creates a sandbox folder in its current location, where it stores all its settings and temporary files. Can be downloaded from the US, UK or Hong Kong.

See: https://xpspeak.software.informer.com/4.1/

5. Unifit

See: https://home.uni-leipzig.de/unifit/downloads.htm

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In Figure 10.9 we examined Nessler’s original method for matching the color of a sample to the color of a standard. Matching the colors was a labor intensive process for the analyst. Not surprisingly, spectroscopic methods of analysis were slow to develop. The 1930s and 1940s saw the introduction of photoelectric transducers for ultraviolet and visible radiation, and thermocouples for infrared radiation. As a result, modern instrumentation for absorption spectroscopy became routinely available in the 1940s—progress has been rapid ever since.

10.3.1 Instrumentation

Frequently an analyst must select—from among several instruments of different design—the one instrument best suited for a particular analysis. In this section we examine several different instruments for molecular absorption spectroscopy, emphasizing their advantages and limitations. Methods of sample introduction are also covered in this section.

Instrument Designs for Molecular UV/Vis Absorption

Filter Photometer. The simplest instrument for molecular UV/Vis absorption is a filter photometer (Figure 10.25), which uses an absorption or interference filter to isolate a band of radiation. The filter is placed between the source and the sample to prevent the sample from decomposing when exposed to higher energy radiation. A filter photometer has a single optical path between the source and detector, and is called a single-beam instrument. The instrument is calibrated to 0% T while using a shutter to block the source radiation from the detector. After opening the shutter, the instrument is calibrated to 100% T using an appropriate blank. The blank is then replaced with the sample and its transmittance measured. Because the source’s incident power and the sensitivity of the detector vary with wavelength, the photometer must be recalibrated whenever the filter is changed. Photometers have the advantage of being relatively inexpensive, rugged, and easy to maintain. Another advantage of a photometer is its portability, making it easy to take into the field. Disadvantages of a photometer include the inability to record an absorption spectrum and the source’s relatively large effective bandwidth, which limits the calibration curve’s linearity.

Note

The percent transmittance varies between 0% and 100%. As we learned in Figure 10.21, we use a blank to determine P₀, which corresponds to 100% T. Even in the absence of light the detector records a signal. Closing the shutter allows us to assign 0% T to this signal. Together, setting 0% T and 100% T calibrates the instrument. The amount of light passing through a sample produces a signal that is greater than or equal to that for 0% T and smaller than or equal to that for 100%T.

Figure 10.25 Schematic diagram of a filter photometer. The analyst either inserts a removable filter or the filters are placed in a carousel, an example of which is shown in the photographic inset. The analyst selects a filter by rotating it into place.

Single-Beam Spectrophotometer. An instrument that uses a monochromator for wavelength selection is called a spectrophotometer. The simplest spectrophotometer is a single-beam instrument equipped with a fixed-wavelength monochromator (Figure 10.26). Single-beam spectrophotometers are calibrated and used in the same manner as a photometer. One example of a single-beam spectrophotometer is Thermo Scientific’s Spectronic 20D+, which is shown in the photographic insert to Figure 10.26. The Spectronic 20D+ has a range of 340–625 nm (950 nm when using a red-sensitive detector), and a fixed effective bandwidth of 20 nm. Battery-operated, hand-held single-beam spectrophotometers are available, which are easy to transport into the field. Other single-beam spectrophotometers also are available with effective bandwidths of 2–8 nm. Fixed wavelength single-beam spectrophotometers are not practical for recording spectra because manually adjusting the wavelength and recalibrating the spectrophotometer is awkward and time-consuming. The accuracy of a single-beam spectrophotometer is limited by the stability of its source and detector over time.

Figure 10.26 Schematic diagram of a fixed-wavelength single-beam spectrophotometer. The photographic inset shows a typical instrument. The shutter remains closed until the sample or blank is placed in the sample compartment. The analyst manually selects the wavelength by adjusting the wavelength dial. Inset photo modified from: Adi (www.commons.wikipedia.org).

Double-Beam Spectrophotometer. The limitations of fixed-wavelength, single-beam spectrophotometers are minimized by using a double-beamspectrophotometer (Figure 10.27). A chopper controls the radiation’s path, alternating it between the sample, the blank, and a shutter. The signal processor uses the chopper’s known speed of rotation to resolve the signal reaching the detector into the transmission of the blank, P₀, and the sample, P_T. By including an opaque surface as a shutter, it is possible to continuously adjust 0% T. The effective bandwidth of a double-beam spectrophotometer is controlled by adjusting the monochromator’s entrance and exit slits. Effective bandwidths of 0.2–3.0 nm are common. A scanning monochromator allows for the automated recording of spectra. Double-beam instruments are more versatile than single-beam instruments, being useful for both quantitative and qualitative analyses, but also are more expensive.

Figure 10.27 Schematic diagram of a scanning, double-beam spectrophotometer. A chopper directs the source’s radiation, using a transparent window to pass radiation to the sample and a mirror to reflect radiation to the blank. The chopper’s opaque surface serves as a shutter, which allows for a constant adjustment of the spectrophotometer’s 0% T. The photographic insert shows a typical instrument. The unit in the middle of the photo is a temperature control unit that allows the sample to be heated or cooled.

Diode Array Spectrometer. An instrument with a single detector can monitor only one wavelength at a time. If we replace a single photomultiplier with many photodiodes, we can use the resulting array of detectors to record an entire spectrum simultaneously in as little as 0.1 s. In a diode array spectrometer the source radiation passes through the sample and is dispersed by a grating (Figure 10.28). The photodiode array is situated at the grating’s focal plane, with each diode recording the radiant power over a narrow range of wavelengths. Because we replace a full monochromator with just a grating, a diode array spectrometer is small and compact.

Figure 10.28 Schematic diagram of a diode array spectrophotometer. The photographic insert shows a typical instrument. Note that the 50-mL beaker provides a sense of scale.

One advantage of a diode array spectrometer is the speed of data acquisition, which allows to collect several spectra for a single sample. Individual spectra are added and averaged to obtain the final spectrum. This signal averaging improves a spectrum’s signal-to-noise ratio. If we add together n spectra, the sum of the signal at any point, x, increases as nS_x, where S_x is the signal. The noise at any point, N_x, is a random event, which increases as √nN_x when we add together nspectra. The signal-to-noise ratio (S/N) after n scans isSN=nSxn−−√Nx=n−−√SxnNx(4.8.1)(4.8.1)SN=nSxnNx=nSxnNx

where S_x/N_x is the signal-to-noise ratio for a single scan. The impact of signal averaging is shown in Figure 10.29. The first spectrum shows the signal for a single scan, which consists of a single, noisy peak. Signal averaging using 4 scans and 16 scans decreases the noise and improves the signal-to-noise ratio. One disadvantage of a photodiode array is that the effective bandwidth per diode is roughly an order of magnitude larger than that for a high quality monochromator.

Figure 10.29 Effect of signal averaging on a spectrum’s signal-to-noise ratio. From top to bottom: spectrum for a single scan; average spectrum after four scans; and average spectrum after adding 16 scans.

Sample Cells. The sample compartment provides a light-tight environment that limits the addition of stray radiation. Samples are normally in the liquid or solution state, and are placed in cells constructed with UV/Vis transparent materials, such as quartz, glass, and plastic (Figure 10.30). A quartz or fused-silica cell is required when working at a wavelength <300 nm where other materials show a significant absorption. The most common pathlength is 1 cm (10 mm), although cells with shorter (as little as 0.1 cm) and longer pathlengths (up to 10 cm) are available. Longer pathlength cells are useful when analyzing a very dilute solution, or for gas samples. The highest quality cells allow the radiation to strike a flat surface at a 90^o angle, minimizing the loss of radiation to reflection. A test tube is often used as a sample cell with simple, single-beam instruments, although differences in the cell’s pathlength and optical properties add an additional source of error to the analysis.

Figure 10.30 Examples of sample cells for UV/Vis spectroscopy. From left to right (with path lengths in parentheses): rectangular plastic cuvette (10.0 mm), rectangular quartz cuvette (5.000 mm), rectangular quartz cuvette (1.000 mm), cylindrical quartz cuvette (10.00 mm), cylindrical quartz cuvette (100.0 mm). Cells often are available as a matched pair, which is important when using a double-beam instrument.

If we need to monitor an analyte’s concentration over time, it may not be possible to physically remove samples for analysis. This is often the case, for example, when monitoring industrial production lines or waste lines, when monitoring a patient’s blood, or when monitoring environmental systems. With a fiber-optic probe we can analyze samples in situ. An example of a remote sensing fiber-optic probe is shown in Figure 10.31. The probe consists of two bundles of fiber-optic cable. One bundle transmits radiation from the source to the probe’s tip, which is designed to allow the sample to flow through the sample cell. Radiation from the source passes through the solution and is reflected back by a mirror. The second bundle of fiber-optic cable transmits the nonabsorbed radiation to the wavelength selector. Another design replaces the flow cell shown in Figure 10.31 with a membrane containing a reagent that reacts with the analyte. When the analyte diffuses across the membrane it reacts with the reagent, producing a product that absorbs UV or visible radiation. The nonabsorbed radiation from the source is reflected or scattered back to the detector. Fiber optic probes that show chemical selectivity are called optrodes.⁶

Figure 10.31 Example of a fiber-optic probe. The inset photographs provide a close-up look at the probe’s flow cell and the reflecting mirror.

Instrument Designs for Infrared Absorption

Filter Photometer. The simplest instrument for IR absorption spectroscopy is a filter photometer similar to that shown in Figure 10.25 for UV/Vis absorption. These instruments have the advantage of portability, and typically are used as dedicated analyzers for gases such as HCN and CO.

Double-beam spectrophotometer. Infrared instruments using a monochromator for wavelength selection use double-beam optics similar to that shown in Figure 10.27. Double-beam optics are preferred over single-beam optics because the sources and detectors for infrared radiation are less stable than those for UV/Vis radiation. In addition, it is easier to correct for the absorption of infrared radiation by atmospheric CO₂ and H₂O vapor when using double-beam optics. Resolutions of 1–3 cm^–1 are typical for most instruments.

Fourier transform spectrometer. In a Fourier transform infrared spectrometer, or FT–IR, the monochromator is replaced with an interferometer (Figure 10.13). Because an FT-IR includes only a single optical path, it is necessary to collect a separate spectrum to compensate for the absorbance of atmospheric CO₂ and H₂O vapor. This is done by collecting a background spectrum without the sample and storing the result in the instrument’s computer memory. The background spectrum is removed from the sample’s spectrum by ratioing the two signals. In comparison to other instrument designs, an FT–IR provides for rapid data acquisition, allowing an enhancement in signal-to-noise ratio through signal-averaging.

Sample Cells. Infrared spectroscopy is routinely used to analyze gas, liquid, and solid samples. Sample cells are made from materials, such as NaCl and KBr, that are transparent to infrared radiation. Gases are analyzed using a cell with a pathlength of approximately 10 cm. Longer pathlengths are obtained by using mirrors to pass the beam of radiation through the sample several times.

A liquid samples may be analyzed using a variety of different sample cells (Figure 10.32). For non-volatile liquids a suitable sample can be prepared by placing a drop of the liquid between two NaCl plates, forming a thin film that typically is less than 0.01 mm thick. Volatile liquids must be placed in a sealed cell to prevent their evaporation.

Figure 10.32 Three examples of IR sample cells: (a) NaCl salts plates; (b) fixed pathlength (0.5 mm) sample cell with NaCl windows; (c) disposable card with a polyethylene window that is IR transparent with the exception of strong absorption bands at 2918 cm^–1 and 2849 cm^–1.

The analysis of solution samples is limited by the solvent’s IR absorbing properties, with CCl₄, CS₂, and CHCl₃ being the most common solvents. Solutions are placed in cells containing two NaCl windows separated by a Teflon spacer. By changing the Teflon spacer, pathlengths from 0.015–1.0 mm can be obtained.

Transparent solid samples can be analyzed directly by placing them in the IR beam. Most solid samples, however, are opaque, and must be dispersed in a more transparent medium before recording the IR spectrum. If a suitable solvent is available, then the solid can be analyzed by preparing a solution and analyzing as described above. When a suitable solvent is not available, solid samples may be analyzed by preparing a mull of the finely powdered sample with a suitable oil. Alternatively, the powdered sample can be mixed with KBr and pressed into an optically transparent pellet.

The analysis of an aqueous sample is complicated by the solubility of the NaCl cell window in water. One approach to obtaining infrared spectra on aqueous solutions is to use attenuated total reflectance instead of transmission. Figure 10.33 shows a diagram of a typical attenuated total reflectance (ATR) FT–IR instrument. The ATR cell consists of a high refractive index material, such as ZnSe or diamond, sandwiched between a low refractive index substrate and a lower refractive index sample. Radiation from the source enters the ATR crystal where it undergoes a series of total internal reflections before exiting the crystal. During each reflection the radiation penetrates into the sample to a depth of a few microns. The result is a selective attenuation of the radiation at those wavelengths where the sample absorbs. ATR spectra are similar, but not identical, to those obtained by measuring the transmission of radiation.

Figure 10.33 FT-IR spectrometer equipped with a diamond ATR sample cell. The inserts show a close-up photo of the sample platform, a sketch of the ATR’s sample slot, and a schematic showing how the source’s radiation interacts with the sample. The pressure tower is used to ensure the contact of solid samples with the ATR crystal.

Solid samples also can be analyzed using an ATR sample cell. After placing the solid in the sample slot, a compression tip ensures that it is in contact with the ATR crystal. Examples of solids that have been analyzed by ATR include polymers, fibers, fabrics, powders, and biological tissue samples. Another reflectance method is diffuse reflectance, in which radiation is reflected from a rough surface, such as a powder. Powdered samples are mixed with a non-absorbing material, such as powdered KBr, and the reflected light is collected and analyzed. As with ATR, the resulting spectrum is similar to that obtained by conventional transmission methods.

Note

Further details about these, and other methods for preparing solids for infrared analysis can be found in this chapter’s additional resources.

10.3.2 Quantitative Applications

The determination of an analyte’s concentration based on its absorption of ultraviolet or visible radiation is one of the most frequently encountered quantitative analytical methods. One reason for its popularity is that many organic and inorganic compounds have strong absorption bands in the UV/Vis region of the electromagnetic spectrum. In addition, if an analyte does not absorb UV/Vis radiation—or if its absorbance is too weak—we often can react it with another species that is strongly absorbing. For example, a dilute solution of Fe²⁺ does not absorb visible light. Reacting Fe²⁺ with o-phenanthroline, however, forms an orange–red complex of Fe(phen)₃²⁺ that has a strong, broad absorbance band near 500 nm. An additional advantage to UV/Vis absorption is that in most cases it is relatively easy to adjust experimental and instrumental conditions so that Beer’s law is obeyed.

Note

Figure 10.18 shows the visible spectrum for Fe(phen)₃²⁺.

A quantitative analysis based on the absorption of infrared radiation, although important, is less frequently encountered than those for UV/Vis absorption. One reason is the greater tendency for instrumental deviations from Beer’s law when using infrared radiation. Because an infrared absorption band is relatively narrow, any deviation due to the lack of monochromatic radiation is more pronounced. In addition, infrared sources are less intense than UV/Vis sources, making stray radiation more of a problem. Differences in pathlength for samples and standards when using thin liquid films or KBr pellets are a problem, although an internal standard can be used to correct for any difference in pathlength. Finally, establishing a 100% T (A = 0) baseline is often difficult because the optical properties of NaCl sample cells may change significantly with wavelength due to contamination and degradation. We can minimize this problem by measuring absorbance relative to a baseline established for the absorption band. Figure 10.34 shows how this is accomplished.

Note

Another approach is to use a cell with a fixed pathlength, such as that shown in Figure 10.32b.

Figure 10.34 Method for determining absorbance from an IR spectrum.

Environmental Applications

The analysis of waters and wastewaters often relies on the absorption of ultraviolet and visible radiation. Many of these methods are outlined in Table 10.6. Several of these methods are described here in more detail.

Although the quantitative analysis of metals in waters and wastewaters is accomplished primarily by atomic absorption or atomic emission spectroscopy, many metals also can be analyzed following the formation of a colored metal–ligand complex. One advantage to these spectroscopic methods is that they are easily adapted to the analysis of samples in the field using a filter photometer. One ligand that is used in the analysis of several metals is diphenylthiocarbazone, also known as dithizone. Dithizone is not soluble in water, but when a solution of dithizone in CHCl₃ is shaken with an aqueous solution containing an appropriate metal ion, a colored metal–dithizonate complex forms that is soluble in CHCl₃. The selectivity of dithizone is controlled by adjusting the sample’s pH. For example, Cd²⁺ is extracted from solutions that are made strongly basic with NaOH, Pb²⁺ from solutions that are made basic with an NH₃/NH₄⁺ buffer, and Hg²⁺ from solutions that are slightly acidic.

Note

Atomic absorption is the subject of Section 10.4 and atomic emission is the subject of Section 10.7.

The structure of dithizone is shown to the right. See Chapter 7 for a discussion of extracting metal ions using dithizone.

When chlorine is added to water the portion available for disinfection is called the chlorine residual. There are two forms of chlorine residual. The free chlorine residual includes Cl₂, HOCl, and OCl^–. The combined chlorine residual, which forms from the reaction of NH₃ with HOCl, consists of monochloramine, NH₂Cl, dichloramine, NHCl₂, and trichloramine, NCl₃. Because the free chlorine residual is more efficient at disinfection, there is an interest in methods that can distinguish between the different forms of the total chlorine residual. One such method is the leuco crystal violet method. The free residual chlorine is determined by adding leuco crystal violet to the sample, which instantaneously oxidizes to give a blue colored compound that is monitored at 592 nm. Completing the analysis in less than five minutes prevents a possible interference from the combined chlorine residual. The total chlorine residual (free + combined) is determined by reacting a separate sample with iodide, which reacts with both chlorine residuals to form HOI. When the reaction is complete, leuco crystal violet is added and oxidized by HOI, giving the same blue colored product. The combined chlorine residual is determined by difference.

Note

In Chapter 9 we explored how the total chlorine residual can be determined by a redox titration; see Representative Method 9.3 for further details. The method described here allows us to divide the total chlorine residual into its component parts.

The concentration of fluoride in drinking water may be determined indirectly by its ability to form a complex with zirconium. In the presence of the dye SPADNS, solutions of zirconium form a red colored compound, called a lake, that absorbs at 570 nm. When fluoride is added, the formation of the stable ZrF₆^2– complex causes a portion of the lake to dissociate, decreasing the absorbance. A plot of absorbance versus the concentration of fluoride, therefore, has a negative slope.

Note

SPADNS, which is shown below, is an abbreviation for the sodium salt of 2-(4-sulfophenylazo)-1,8-dihydroxy-3,6-napthalenedisulfonic acid, which is a mouthful to say.

Spectroscopic methods also are used to determine organic constituents in water. For example, the combined concentrations of phenol, and ortho- and meta- substituted phenols are determined by using steam distillation to separate the phenols from nonvolatile impurities. The distillate reacts with 4-aminoantipyrine at pH 7.9 ± 0.1 in the presence of K₃Fe(CN)₆, forming a yellow colored antipyrine dye. After extracting the dye into CHCl₃, its absorbance is monitored at 460 nm. A calibration curve is prepared using only the unsubstituted phenol, C₆H₅OH. Because the molar absorptivity of substituted phenols are generally less than that for phenol, the reported concentration represents the minimum concentration of phenolic compounds.

Molecular absorption also can be used for the analysis of environmentally significant airborne pollutants. In many cases the analysis is carried out by collecting the sample in water, converting the analyte to an aqueous form that can be analyzed by methods such as those described in Table 10.6. For example, the concentration of NO₂ can be determined by oxidizing NO₂ to NO₃^–. The concentration of NO₃^– is then determined by first reducing it to NO₂^– with Cd, and then reacting NO₂^– with sulfanilamide and N-(1-naphthyl)-ethylenediamine to form a red azo dye. Another important application is the analysis for SO₂, which is determined by collecting the sample in an aqueous solution of HgCl₄^2– where it reacts to form Hg(SO₃)₂^2–. Addition of p-rosaniline and formaldehyde produces a purple complex that is monitored at 569 nm. Infrared absorption is useful for the analysis of organic vapors, including HCN, SO₂, nitrobenzene, methyl mercaptan, and vinyl chloride. Frequently, these analyses are accomplished using portable, dedicated infrared photometers.

Clinical Applications

The analysis of clinical samples is often complicated by the complexity of the sample matrix, which may contribute a significant background absorption at the desired wavelength. The determination of serum barbiturates provides one example of how this problem is overcome. The barbiturates are first extracted from a sample of serum with CHCl₃ and then extracted from the CHCl₃ into 0.45 M NaOH (pH ≈ 13). The absorbance of the aqueous extract is measured at 260 nm, and includes contributions from the barbiturates as well as other components extracted from the serum sample. The pH of the sample is then lowered to approximately 10 by adding NH₄Cl and the absorbance remeasured. Because the barbiturates do not absorb at this pH, we can use the absorbance at pH 10, A_pH 10, to correct the absorbance at pH 13, A_pH 13Abarb=ApH 13−Vsamp+VNH4ClVsamp×ApH 10(4.8.2)(4.8.2)Abarb=ApH 13−Vsamp+VNH4ClVsamp×ApH 10

where A_barb is the absorbance due to the serum barbiturates, and V_samp and V_NH4Cl are the volumes of sample and NH₄Cl, respectively. Table 10.7 provides a summary of several other methods for analyzing clinical samples.

Industrial Analysis

UV/Vis molecular absorption is used for the analysis of a diverse array of industrial samples including pharmaceuticals, food, paint, glass, and metals. In many cases the methods are similar to those described in Table 10.6 and Table 10.7. For example, the amount of iron in food can be determined by bringing the iron into solution and analyzing using the o-phenanthroline method listed in Table 10.6.

Many pharmaceutical compounds contain chromophores that make them suitable for analysis by UV/Vis absorption. Products that have been analyzed in this fashion include antibiotics, hormones, vitamins, and analgesics. One example of the use of UV absorption is in determining the purity of aspirin tablets, for which the active ingredient is acetylsalicylic acid. Salicylic acid, which is produced by the hydrolysis of acetylsalicylic acid, is an undesirable impurity in aspirin tablets, and should not be present at more than 0.01% w/w. Samples can be screened for unacceptable levels of salicylic acid by monitoring the absorbance at a wavelength of 312 nm. Acetylsalicylic acid absorbs at 280 nm, but absorbs poorly at 312 nm. Conditions for preparing the sample are chosen such that an absorbance of greater than 0.02 signifies an unacceptable level of salicylic acid.

Forensic Applications

UV/Vis molecular absorption is routinely used for the analysis of narcotics and for drug testing. One interesting forensic application is the determination of blood alcohol using the Breathalyzer test. In this test a 52.5-mL breath sample is bubbled through an acidified solution of K₂Cr₂O₇, which oxidizes ethanol to acetic acid. The concentration of ethanol in the breath sample is determined by the decrease in absorbance at 440 nm where the dichromate ion absorbs. A blood alcohol content of 0.10%, which is above the legal limit, corresponds to 0.025 mg of ethanol in the breath sample.

Developing a Quantitative Method for a Single Component

In developing a quantitative analytical method, the conditions under which Beer’s law is obeyed must be established. First, the most appropriate wavelength for the analysis is determined from an absorption spectrum. In most cases the best wavelength corresponds to an absorption maximum because it provides greater sensitivity and is less susceptible to instrumental limitations. Second, if an instrument with adjustable slits is being used, then an appropriate slit width needs to be chosen. The absorption spectrum also aids in selecting a slit width. Usually we set the slits to be as wide as possible because this increases the throughput of source radiation, while also being narrow enough to avoid instrumental limitations to Beer’s law. Finally, a calibration curve is constructed to determine the range of concentrations for which Beer’s law is valid. Additional considerations that are important in any quantitative method are the effect of potential interferents and establishing an appropriate blank.

Note

The best way to appreciate the theoretical and practical details discussed in this section is to carefully examine a typical analytical method. Although each method is unique, the following description of the determination of iron in water and wastewater provides an instructive example of a typical procedure. The description here is based on Method 3500- Fe B as published in Standard Methods for the Examination of Water and Wastewater, 20th Ed., American Public Health Association: Washington, D. C., 1998.

Representative Method 10.1

Determination of Iron in Water and Wastewater

Description of Method

Iron in the +2 oxidation state reacts with o-phenanthroline to form the orange-red Fe(phen)₃²⁺ complex. The intensity of the complex’s color is independent of solution acidity between a pH of 3 and 9. Because the complex forms more rapidly at lower pH levels, the reaction is usually carried out within a pH range of 3.0–3.5. Any iron present in the +3 oxidation state is reduced with hydroxylamine before adding o-phenanthroline. The most important interferents are strong oxidizing agents, polyphosphates, and metal ions such as Cu²⁺, Zn²⁺, Ni²⁺, and Cd²⁺. An interference from oxidizing agents is minimized by adding an excess of hydroxylamine, and an interference from polyphosphate is minimized by boiling the sample in the presence of acid. The absorbance of samples and standards are measured at a wavelength of 510 nm using a 1-cm cell (longer pathlength cells also may be used). Beer’s law is obeyed for concentrations of within the range of 0.2–4.0 mg Fe/L.

(Figure 10.18 shows the visible spectrum for Fe(phen)₃²⁺.)

Procedure

For samples containing less than 2 mg Fe/L, directly transfer a 50-mL portion to a 125-mL Erlenmeyer flask. Samples containing more than 2 mg Fe/L must be diluted before acquiring the 50-mL portion. Add 2 mL of concentrated HCl and 1 mL of hydroxylamine to the sample. Heat the solution to boiling and continue boiling until the solution’s volume is reduced to between 15 and 20 mL. After cooling to room temperature, transfer the solution to a 50-mL volumetric flask, add 10 mL of an ammonium acetate buffer, 2 mL of a 1000 ppm solution of o-phenanthroline, and dilute to volume. Allow 10–15 minutes for color development before measuring the absorbance, using distilled water to set 100% T. Calibration standards, including a blank, are prepared by the same procedure using a stock solution containing a known concentration of Fe²⁺.

Questions

1. Explain why strong oxidizing agents are interferents, and why an excess of hydroxylamine prevents the interference.

A strong oxidizing agent oxidizes some Fe²⁺ to Fe³⁺. Because Fe(phen)₃³⁺ does not absorb as strongly as Fe(phen)₃²⁺, the absorbance decreases, producing a negative determinate error. The excess hydroxylamine reacts with the oxidizing agents, removing them from the solution.

2. The color of the complex is stable between pH levels of 3 and 9. What are some possible complications at more acidic or more basic pH’s?

Because o-phenanthroline is a weak base, its conditional formation constant for Fe(phen)₃²⁺ is less favorable at more acidic pH levels, where o-phenanthroline is protonated. The result is a decrease in absorbance and a less sensitive analytical method.

(In Chapter 9 we saw the same effect of pH on the complexation reactions between EDTA and metal ions.)

When the pH is greater than 9, competition between OH^– and o-phenanthroline for Fe²⁺ also decreased the absorbance. In addition, if the pH is sufficiently basic there is a risk that the iron will precipitate as Fe(OH)₂.

3. Cadmium is an interferent because it forms a precipitate with o-phenanthroline. What effect would the formation of precipitate have on the determination of iron?

Because o-phenanthroline is present in large excess (2000 μg of o-phenanthroline for 100 μg of Fe²⁺), it is not likely that the interference is due to an insufficient amount of o-phenanthroline being available to react with the Fe²⁺. The presence of a precipitate in the sample cell results in the scattering of radiation, which causes an apparent increase in absorbance. Because the measured absorbance increases, the reported concentration is too high.

(Although scattering is a problem here, it can serve as the basis of a useful analytical method. See Section 10.8 for further details.)

4. Even high quality ammonium acetate contains a significant amount of iron. Why is this source of iron not a problem?

Because all samples and standards are prepared using the same volume of ammonium acetate buffer, the contribution of this source of iron is accounted for by the calibration curve’s reagent blank.

Quantitative Analysis for a Single Analyte

To determine the concentration of a an analyte we measure its absorbance and apply Beer’s law using any of the standardization methods described in Chapter 5. The most common methods are a normal calibration curve using external standards and the method of standard additions. A single point standardization is also possible, although we must first verify that Beer’s law holds for the concentration of analyte in the samples and the standard.

Example 10.5

The determination of Fe in an industrial waste stream was carried out by the o‑phenanthroline described in Representative Method 10.1. Using the data in the following table, determine the mg Fe/L in the waste stream.

Solution

Linear regression of absorbance versus the concentration of Fe in the standards gives a calibration curve with the following equation.A=0.0006+0.1817×(mgFe/L)(4.8.3)(4.8.3)A=0.0006+0.1817×(mgFe/L)

Substituting the sample’s absorbance into the calibration expression gives the concentration of Fe in the waste stream as 1.48 mg Fe/L.

Practice Exercise 10.5

The concentration of Cu²⁺ in a sample can be determined by reacting it with the ligand cuprizone and measuring its absorbance at 606 nm in a 1.00-cm cell. When a 5.00-mL sample is treated with cuprizone and diluted to 10.00 mL, the resulting solution has an absorbance of 0.118. A second 5.00-mL sample is mixed with 1.00 mL of a 20.00 mg/L standard of Cu²⁺, treated with cuprizone and diluted to 10.00 mL, giving an absorbance of 0.162. Report the mg Cu²⁺/L in the sample.

Click here to review your answer to this exercise.

Quantitative Analysis of Mixtures

Suppose we need to determine the concentration of two analytes, X and Y, in a sample. If each analyte has a wavelength where the other analyte does not absorb, then we can proceed using the approach in Example 10.5. Unfortunately, UV/Vis absorption bands are so broad that it frequently is not possible to find suitable wavelengths. Because Beer’s law is additive the mixture’s absorbance, A_mix, is(Amix)λ1=(εX)λ1bCX+(εY)λ1bCY(10.11)(10.11)(Amix)λ1=(εX)λ1bCX+(εY)λ1bCY

where λ1 is the wavelength at which we measure the absorbance. Because equation 10.11 includes terms for the concentration of both X and Y, the absorbance at one wavelength does not provide enough information to determine either C_X or C_Y. If we measure the absorbance at a second wavelength(Amix)λ2=(εX)λ2bCX+(εY)λ2bCY(10.12)(10.12)(Amix)λ2=(εX)λ2bCX+(εY)λ2bCY

then C_X and C_Y can be determined by solving simultaneously equation10.11 and equation 10.12. Of course, we also must determine the value for ε_X and ε_Y at each wavelength. For a mixture of n components, we must measure the absorbance at n different wavelengths.

Example 10.6

The concentrations of Fe³⁺ and Cu²⁺ in a mixture can be determined following their reaction with hexacyanoruthenate (II), Ru(CN)₆^4–, which forms a purple-blue complex with Fe³⁺ (λ_max = 550 nm) and a pale-green complex with Cu²⁺ (λ_max = 396 nm).⁷ The molar absorptivities (M^–1 cm^–1) for the metal complexes at the two wavelengths are summarized in the following table.

When a sample containing Fe³⁺ and Cu²⁺ is analyzed in a cell with a pathlength of 1.00 cm, the absorbance at 550 nm is 0.183 and the absorbance at 396 nm is 0.109. What are the molar concentrations of Fe³⁺ and Cu²⁺ in the sample?

Solution

Substituting known values into equations 10.11 and 10.12 givesA550=0.183=9970CFe+34CCu(4.8.4)(4.8.4)A550=0.183=9970CFe+34CCuA396=0.109=84CFe+856CCu(4.8.5)(4.8.5)A396=0.109=84CFe+856CCu

To determine C_Fe and C_Cu we solve the first equation for C_CuCCu=0.183–9970CFe34(4.8.6)(4.8.6)CCu=0.183–9970CFe34

and substitute the result into the second equation.0.109=84CFe+856×0.183−9970CFe34=4.607–(2.51×105)CFe(4.8.7)(4.8.7)0.109=84CFe+856×0.183−9970CFe34=4.607–(2.51×105)CFe

Solving for C_Fe gives the concentration of Fe³⁺ as 1.79 × 10^–5 M. Substituting this concentration back into the equation for the mixture’s absorbance at 396 nm gives the concentration of Cu²⁺ as 1.26 × 10^–4 M.

(Another approach is to multiply the first equation by 856/34 giving4.607=251009CFe+856CCu(4.8.8)(4.8.8)4.607=251009CFe+856CCu

Subtracting the second equation from this equation4.607=251009CFe+856CCu−0.109=84CFe+856CCu–––––––––––––––––––––––––––––––4.498=250925CFe(4.8.9)(4.8.10)(4.8.11)(4.8.9)−4.607=251009CFe+856CCu(4.8.10)−0.109=84CFe+856CCu6CCu_(4.8.11)−4.498=250925CFe

we find that C_Fe is 1.79×10^–5. Having determined C_Fe we can substitute back into one of the other equations to solve for C_Cu, which is 1.26×10^–5.)

Practice Exercise 10.6

The absorbance spectra for Cr³⁺ and Co²⁺ overlap significantly. To determine the concentration of these analytes in a mixture, its absorbance was measured at 400 nm and at 505 nm, yielding values of 0.336 and 0.187, respectively. The individual molar absorptivities (M^–1 cm^–1) are

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To obtain results with good accuracy and precision the two wavelengths should be selected so that ε_X > ε_Y at one wavelength and ε_X < ε_Y at the other wavelength. It is easy to appreciate why this is true. Because the absorbance at each wavelength is dominated by one analyte, any uncertainty in the concentration of the other analyte has less of an impact. Figure 10.35 shows that the choice of wavelengths for Practice Exercise 10.6 are reasonable. When the choice of wavelengths is not obvious, one method for locating the optimum wavelengths is to plot ε_X/ε_Y as function of wavelength, and determine the wavelengths where ε_X/ε_Y reaches maximum and minimum values.⁸

Note

For example, in Example 10.6 the molar absorptivity for Fe³⁺ at 550 nm is 119× that for Cu²⁺, and the molar absorptivity for Cu²⁺ at 396 nm is 10.2× that for Fe³⁺.

Figure 10.35 Visible absorption spectra for 0.0250 M Cr³⁺, 0.0750 M Co²⁺, and for a mixture of Cr³⁺ and Co²⁺. The two wavelengths used for analyzing the mixture of Cr³⁺ and Co²⁺ are shown by the dashed lines. The data for the two standard solutions are from reference 7.

When the analyte’s spectra overlap severely, such that ε_X ≈ ε_Y at all wavelength, other computational methods may provide better accuracy and precision. In a multiwavelength linear regression analysis, for example, a mixture’s absorbance is compared to that for a set of standard solutions at several wavelengths.⁹ If A_SXand A_SY are the absorbance values for standard solutions of components X and Y at any wavelength, thenASX=εXbCSX(10.13)(10.13)ASX=εXbCSXASY=εYbCSY(10.14)(10.14)ASY=εYbCSY

where C_SX and C_SY are the known concentrations of X and Y in the standard solutions. Solving equation 10.13 and equation 10.14 for ε_X and ε_Y, substituting into equation 10.11, and rearranging, givesAmixASX=CXCSX+CYCSY×ASYASX(4.8.12)(4.8.12)AmixASX=CXCSX+CYCSY×ASYASX

To determine C_X and C_Y the mixture’s absorbance and the absorbances of the standard solutions are measured at several wavelengths. Graphing A_mix/A_SX versus A_SY/A_SX gives a straight line with a slope of C_Y/C_SY and a y-intercept of C_X/C_SX. This approach is particularly helpful when it is not possible to find wavelengths where ε_X > ε_Y and ε_X < ε_Y.

Note

The approach outlined here for a multiwavelength linear regression uses a single standard solution for each analyte. A more rigorous approach uses multiple standards for each analyte. The math behind the analysis of this data—what we call a multiple linear regression—is beyond the level of this text. For more details about multiple linear regression see Brereton, R. G. Chemometrics: Data Analysis for the Laboratory and Chemical Plant, Wiley: Chichester, England, 2003.

Example 10.7

Figure 10.35 shows visible absorbance spectra for a standard solution of 0.0250 M Cr³⁺, a standard solution of 0.0750 M Co²⁺, and a mixture containing unknown concentrations of each ion. The data for these spectra are shown here.¹⁰

Use a multiwavelength regression analysis to determine the composition of the unknown.

Solution

First we need to calculate values for A_mix/A_SX and for A_SY/A_SX. Let’s define X as Co²⁺ and Y as Cr³⁺. For example, at a wavelength of 375 nm A_mix/A_SX is 0.53/0.01, or 53 and A_SY/A_SX is 0.26/0.01, or 26. Completing the calculation for all wavelengths and graphing A_mix/A_SX versus A_SY/A_SXgives the result shown in Figure 10.36. Fitting a straight-line to the data gives a regression model ofAmixASX=0.636+2.01×ASYASX(4.8.13)(4.8.13)AmixASX=0.636+2.01×ASYASX

Using the y-intercept, the concentration of Co²⁺ isCXCSX=CCo0.0750M=0.636(4.8.14)(4.8.14)CXCSX=CCo0.0750M=0.636

or C_Co = 0.048 M, and using the slope the concentration of Cr³⁺ isCYCSY=CCr0.0250M=2.01(4.8.15)(4.8.15)CYCSY=CCr0.0250M=2.01

or C_Cr = 0.050 M.

Figure 10.36 Multiwavelength linear regression analysis for the data in Example 10.7.

Practice Exercise 10.7

A mixture of MnO₄^– and Cr₂O₇^2–, and standards of 0.10 mM KMnO₄ and of 0.10 mM K₂Cr₂O₇ gives the results shown in the following table. Determine the composition of the mixture. The data for this problem is from Blanco, M. C.; Iturriaga, H.; Maspoch, S.; Tarin, P. J. Chem. Educ. 1989, 66, 178–180.

(There are many additional ways to analyze mixtures spectrophotometrically, including generalized standard additions, H-point standard additions, principal component regression to name a few. Consult the chapter’s additional resources for further information.)

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10.3.3 Qualitative Applications

As discussed earlier in Section 10.2.1, ultraviolet, visible, and infrared absorption bands result from the absorption of electromagnetic radiation by specific valence electrons or bonds. The energy at which the absorption occurs, and its intensity is determined by the chemical environment of the absorbing moiety. For example, benzene has several ultraviolet absorption bands due to π → π* transitions. The position and intensity of two of these bands, 203.5 nm (ε = 7400 M^–1 cm^–1) and 254 nm (ε = 204 M^–1 cm^–1), are sensitive to substitution. For benzoic acid, in which a carboxylic acid group replaces one of the aromatic hydrogens, the two bands shift to 230 nm (ε = 11 600 M^–1 cm^–1) and 273 nm (ε = 970 M^–1 cm^–1). A variety of rules have been developed to aid in correlating UV/Vis absorption bands to chemical structure. Similar correlations have been developed for infrared absorption bands. For example a carbonyl’s C=O stretch is sensitive to adjacent functional groups, occurring at 1650 cm^–1 for acids, 1700 cm^–1 for ketones, and 1800 cm^–1 for acid chlorides. The interpretation of UV/Vis and IR spectra receives adequate coverage elsewhere in the chemistry curriculum, notably in organic chemistry, and is not considered further in this text.

With the availability of computerized data acquisition and storage it is possible to build digital libraries of standard reference spectra. The identity of an a unknown compound can often be determined by comparing its spectrum against a library of reference spectra, a process is known as spectral searching. Comparisons are made using an algorithm that calculates the cumulative difference between the sample’s spectrum and a reference spectrum. For example, one simple algorithm uses the following equationD=∑i=1n|(Asample)i−(Areference)i|(4.8.16)(4.8.16)D=∑i=1n|(Asample)i−(Areference)i|

where D is the cumulative difference, A_sample is the sample’s absorbance at wavelength or wavenumber i, A_reference is the absorbance of the reference compound at the same wavelength or wavenumber, and n is the number of digitized points in the spectra. The cumulative difference is calculated for each reference spectrum. The reference compound with the smallest value of D provides the closest match to the unknown compound. The accuracy of spectral searching is limited by the number and type of compounds included in the library, and by the effect of the sample’s matrix on the spectrum.

Another advantage of computerized data acquisition is the ability to subtract one spectrum from another. When coupled with spectral searching it may be possible, by repeatedly searching and subtracting reference spectra, to determine the identity of several components in a sample without the need of a prior separation step. An example is shown in Figure 10.37 in which the composition of a two-component mixture is determined by successive searching and subtraction. Figure 10.37a shows the spectrum of the mixture. A search of the spectral library selects cocaine ^. HCl (Figure 10.37b) as a likely component of the mixture. Subtracting the reference spectrum for cocaine ^. HCl from the mixture’s spectrum leaves a result (Figure 10.37c) that closely matches mannitol’s reference spectrum (Figure 10.37d). Subtracting the reference spectrum for leaves only a small residual signal (Figure 10.37e).

Figure 10.37 Identifying the components of a mixture by spectral searching and subtracting. (a) IR spectrum of the mixture; (b) Reference IR spectrum of cocaine^. HCl; (c) Result of subtracting the spectrum of cocaine ^. HCl from the mixture’s spectrum; (d) Reference IR spectrum of mannitol; and (e) The residual spectrum after removing mannitol’s contribution to the mixture’s spectrum.

Note

IR spectra traditionally are displayed using percent transmittance, %T, along the y-axis (for example, see Figure 10.16). Because absorbance—not percent transmittance—is a linear function of concentration, spectral searching and spectral subtraction, is easier to do when displaying absorbance on the y-axis.

10.3.4 Characterization Applications

Molecular absorption, particularly in the UV/Vis range, has been used for a variety of different characterization studies, including determining the stoichiometry of metal–ligand complexes and determining equilibrium constants. Both of these examples are examined in this section.

Stoichiometry of a Metal-Ligand Complex

We can determine the stoichiometry of a metal–ligand complexation reactionM+yL⇋MLy(4.8.17)(4.8.17)M+yL⇋MLy

using one of three methods: the method of continuous variations, the mole-ratio method, and the slope-ratio method. Of these approaches, the method of continuous variations, also called Job’s method, is the most popular. In this method a series of solutions is prepared such that the total moles of metal and ligand, n_total, in each solution is the same. If (n_M)_i and (n_L)_i are, respectively, the moles of metal and ligand in solution i, thenntotal=(nM)i+(nL)i(4.8.18)(4.8.18)ntotal=(nM)i+(nL)i

The relative amount of ligand and metal in each solution is expressed as the mole fraction of ligand, (X_L)_i, and the mole fraction of metal, (X_M)_i,(XL)i=(nL)intotal(4.8.19)(4.8.19)(XL)i=(nL)intotal(XM)i=1−(nL)intotal=(nM)intotal(4.8.20)(4.8.20)(XM)i=1−(nL)intotal=(nM)intotal

The concentration of the metal–ligand complex in any solution is determined by the limiting reagent, with the greatest concentration occurring when the metal and the ligand are mixed stoichiometrically. If we monitor the complexation reaction at a wavelength where only the metal–ligand complex absorbs, a graph of absorbance versus the mole fraction of ligand will have two linear branches—one when the ligand is the limiting reagent and a second when the metal is the limiting reagent. The intersection of these two branches represents a stoichiometric mixing of the metal and the ligand. We can use the mole fraction of ligand at the intersection to determine the value of y for the metal–ligand complex ML_y.y=nLnM=XLXM=XL1−XL(4.8.21)(4.8.21)y=nLnM=XLXM=XL1−XL

Note

You also can plot the data as absorbance versus the mole fraction of metal. In this case, y is equal to (1–X_M)/X_M.

Example 10.8

To determine the formula for the complex between Fe²⁺ and o-phenanthroline, a series of solutions is prepared in which the total concentration of metal and ligand is held constant at 3.15 × 10^–4 M. The absorbance of each solution is measured at a wavelength of 510 nm. Using the following data, determine the formula for the complex.

(To prepare the solutions for this example I first prepared a solution of 3.15 × 10^-4 M Fe²⁺ and a solution of 3.15 × 10^-4 M o-phenanthroline. Because the two stock solutions are of equal concentration, diluting a portion of one solution with the other solution gives a mixture in which the combined concentration of o-phenanthroline and Fe²⁺ is 3.15 × 10^-4 M. If each solution has the same volume, then each solution contains the same total moles of metal and ligand.)

Solution

A plot of absorbance versus the mole fraction of ligand is shown in Figure 10.38. To find the maximum absorbance, we extrapolate the two linear portions of the plot. The two lines intersect at a mole fraction of ligand of 0.75. Solving for y givesy=XL1–XL=0.751–0.75=3(4.8.22)(4.8.22)y=XL1–XL=0.751–0.75=3

The formula for the metal–ligand complex is Fe(o-phenanthroline)₃²⁺.

Figure 10.38 Continuous variations plot for Example 10.8. The photo shows the solutions used in gathering the data. Each solution is displayed directly below its corresponding point on the continuous variations plot.

Practice Exercise 10.8

Use the continuous variations data in the following table to determine the formula for the complex between Fe²⁺ and SCN^–. The data for this problem is adapted from Meloun, M.; Havel, J.; Högfeldt, E. Computation of Solution Equilibria, Ellis Horwood: Chichester, England, 1988, p. 236.

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Several precautions are necessary when using the method of continuous variations. First, the metal and the ligand must form only one metal–ligand complex. To determine if this condition is true, plots of absorbance versus X_L are constructed at several different wavelengths and for several different values of n_total. If the maximum absorbance does not occur at the same value of X_L for each set of conditions, then more than one metal–ligand complex must be present. A second precaution is that the metal–ligand complex’s absorbance must obey Beer’s law. Third, if the metal–ligand complex’s formation constant is relatively small, a plot of absorbance versus X_L may show significant curvature. In this case it is often difficult to determine the stoichiometry by extrapolation. Finally, because the stability of a metal–ligand complex may be influenced by solution conditions, the composition of the solutions must be carefully controlled. When the ligand is a weak base, for example, the solutions must be buffered to the same pH.

In the mole-ratio method the amount of one reactant, usually the moles of metal, is held constant, while the amount of the other reactant is varied. The absorbance is monitored at a wavelength where the metal–ligand complex absorbs. A plot of absorbance as a function of the ligand-to-metal mole ratio, n_L/n_M, has two linear branches, which intersect at a mole–ratio corresponding to the complex’s formula. Figure 10.39a shows a mole-ratio plot for the formation of a 1:1 complex in which the absorbance is monitored at a wavelength where only the complex absorbs. Figure 10.39b shows a mole-ratio plot for a 1:2 complex in which all three species—the metal, the ligand, and the complex—absorb at the selected wavelength. Unlike the method of continuous variations, the mole-ratio method can be used for complexation reactions that occur in a stepwise fashion if there is a difference in the molar absorptivities of the metal–ligand complexes, and if the formation constants are sufficiently different. A typical mole-ratio plot for the step-wise formation of ML and ML₂ is shown in Figure 10.39c.

Figure 10.39 Mole-ratio plots for: (a) a 1:1 metal–ligand complex in which only the complex absorbs; (b) a 1:2 metal–ligand complex in which the metal, the ligand, and the complex absorb; and (c) the stepwise formation of a 1:1 and a 1:2 metal–ligand complex.

In both the method of continuous variations and the mole-ratio method we determine the complex’s stoichiometry by extrapolating absorbance data from conditions in which there is a linear relationship between absorbance and the relative amounts of metal and ligand. If a metal–ligand complex is very weak, a plot of absorbance versus X_L or n_L/n_M may be so curved that it is impossible to determine the stoichiometry by extrapolation. In this case the slope-ratio may be used.

In the slope-ratio method two sets of solutions are prepared. The first set of solutions contains a constant amount of metal and a variable amount of ligand, chosen such that the total concentration of metal, C_M, is much larger than the total concentration of ligand, C_L. Under these conditions we may assume that essentially all the ligand reacts in forming the metal–ligand complex. The concentration of the complex, which has the general form M_xL_y, is[MxLy]=CLy(4.8.23)(4.8.23)[MxLy]=CLy

If we monitor the absorbance at a wavelength where only M_xL_y absorbs, thenA=εb[MxLy]=εbCLy(4.8.24)(4.8.24)A=εb[MxLy]=εbCLy

and a plot of absorbance versus C_L is linear with a slope, s_L, ofsL=εby(4.8.25)(4.8.25)sL=εby

A second set of solutions is prepared with a fixed concentration of ligand that is much greater than a variable concentration of metal; thus[MxLy]=CMx(4.8.26)(4.8.26)[MxLy]=CMxA=εb[MxLy]=εbCMx(4.8.27)(4.8.27)A=εb[MxLy]=εbCMxsM=εbx(4.8.28)(4.8.28)sM=εbx

A ratio of the slopes provides the relative values of x and y.sMsL=εb/xεb/y=yx(4.8.29)(4.8.29)sMsL=εb/xεb/y=yx

An important assumption in the slope-ratio method is that the complexation reaction continues to completion in the presence of a sufficiently large excess of metal or ligand. The slope-ratio method also is limited to systems in which only a single complex is formed and for which Beer’s law is obeyed.

Determination of Equilibrium Constants

Another important application of molecular absorption spectroscopy is the determination of equilibrium constants. Let’s consider, as a simple example, an acid–base reaction of the general formHIn(aq)+H2O(l)⇋H3O+(aq)+In−(aq)(4.8.30)(4.8.30)HIn(aq)+H2O(l)⇋H3O+(aq)+In−(aq)

where HIn and In^– are the conjugate weak acid and weak base forms of an acid–base indicator. The equilibrium constant for this reaction isKa=[H3O+][In−][HIn](4.8.31)(4.8.31)Ka=[H3O+][In−][HIn]

To determine the equilibrium constant’s value, we prepare a solution in which the reaction is in a state of equilibrium and determine the equilibrium concentration of H₃O⁺, HIn, and In^–. The concentration of H₃O⁺ is easy to determine by simply measuring the solution’s pH. To determine the concentration of HIn and In^– we can measure the solution’s absorbance.

If both HIn and In^– absorb at the selected wavelength, then, from equation 10.6, we know thatA=εHInb[HIn]+εInb[In−](10.15)(10.15)A=εHInb[HIn]+εInb[In−]

where ε_HIn and ε_In are the molar absorptivities for HIn and In^–. The total concentration of indicator, C, is given by a mass balance equationC=[HIn]+[In−](10.16)(10.16)C=[HIn]+[In−]

Solving equation 10.16 for [HIn] and substituting into equation 10.15 givesA=εHInb(C−[In−])+εInb[In−](4.8.32)(4.8.32)A=εHInb(C−[In−])+εInb[In−]

which we simplify toA=εHInbC−εHInb[In−]+εInb[In−](4.8.33)(4.8.33)A=εHInbC−εHInb[In−]+εInb[In−]A=AHIn+b[In−](εIn−εHIn)(10.17)(10.17)A=AHIn+b[In−](εIn−εHIn)

where A_HIn, which is equal to ε_HInbC, is the absorbance when the pH is acidic enough that essentially all the indicator is present as HIn. Solving equation 10.17 for the concentration of In^– gives[In−]=A−AHInb(εIn−εHIn)(10.18)(10.18)[In−]=A−AHInb(εIn−εHIn)

Proceeding in the same fashion, we can derive a similar equation for the concentration of HIn[HIn]=AIn−Ab(εIn−εHIn)(10.19)(10.19)[HIn]=AIn−Ab(εIn−εHIn)

where A_In, which is equal to ε_InbC, is the absorbance when the pH is basic enough that only In^– contributes to the absorbance. Substituting equation 10.18 and equation 10.19 into the equilibrium constant expression for HIn givesKa=[H3O+]A−AHInAIn−A(10.20)(10.20)Ka=[H3O+]A−AHInAIn−A

We can use equation 10.20 to determine the value of K_a in one of two ways. The simplest approach is to prepare three solutions, each of which contains the same amount, C, of indicator. The pH of one solution is made sufficiently acidic such that [HIn] >> [In⁻]. The absorbance of this solution gives A_HIn. The value of A_In is determined by adjusting the pH of the second solution such that [In⁻] >> [HIn]. Finally, the pH of the third solution is adjusted to an intermediate value, and the pH and absorbance, A, recorded. The value of K_a is calculated using equation 10.20.

Example 10.9

The acidity constant for an acid–base indicator is determined by preparing three solutions, each of which has a total indicator concentration of 5.00 × 10^–5 M. The first solution is made strongly acidic with HCl and has an absorbance of 0.250. The second solution was made strongly basic and has an absorbance of 1.40. The pH of the third solution is 2.91 and has an absorbance of 0.662. What is the value of K_a for the indicator?

Solution

The value of K_a is determined by making appropriate substitutions into 10.20; thusKa=(1.23×10−3)×0.662−0.2501.40−0.662=6.87×10−4(4.8.34)(4.8.34)Ka=(1.23×10−3)×0.662−0.2501.40−0.662=6.87×10−4

Practice Exercise 10.9

To determine the K_a of a merocyanine dye, the absorbance of a solution of 3.5×10^–4 M dye was measured at a pH of 2.00, a pH of 6.00, and a pH of 12.00, yielding absorbances of 0.000, 0.225, and 0.680, respectively. What is the value of K_a for this dye? The data for this problem is adapted from Lu, H.; Rutan, S. C. Anal. Chem., 1996, 68, 1381–1386.

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A second approach for determining K_a is to prepare a series of solutions, each containing the same amount of indicator. Two solutions are used to determine values for A_HIn and A_In. Taking the log of both sides of equation 10.20 and rearranging leave us with the following equation.logA−AHInAIn−A=pH−pKa(10.21)(10.21)log⁡A−AHInAIn−A=pH−pKa

A plot of log[(A – A_HIn)/(A_In – A)] versus pH is a straight-line with a slope of +1 and a y-intercept of –pK_a.

Practice Exercise 10.10

To determine the K_a of the indicator bromothymol blue, the absorbance of a series of solutions containing the same concentration of the indicator was measured at pH levels of 3.35, 3.65, 3.94, 4.30, and 4.64, yielding absorbances of 0.170, 0.287, 0.411, 0.562, and 0.670, respectively. Acidifying the first solution to a pH of 2 changes its absorbance to 0.006, and adjusting the pH of the last solution to 12 changes its absorbance to 0.818. What is the value of K_a for this day? The data for this problem is from Patterson, G. S. J. Chem. Educ., 1999, 76, 395–398.

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In developing these approaches for determining K_a we considered a relatively simple system in which the absorbance of HIn and In^– are easy to measure and for which it is easy to determine the concentration of H₃O⁺. In addition to acid–base reactions, we can adapt these approaches to any reaction of the general formX(aq)+Y(aq)⇋Z(aq)(4.8.35)(4.8.35)X(aq)+Y(aq)⇋Z(aq)

including metal–ligand complexation reactions and redox reactions, provided that we can determine spectrophotometrically the concentration of the product, Z, and one of the reactants, and that the concentration of the other reactant can be measured by another method. With appropriate modifications, more complicated systems, in which one or more of these parameters can not be measured, also can be treated.¹¹

10.3.5 Evaluation of UV/Vis and IR Spectroscopy

Scale of Operation

Molecular UV/Vis absorption is routinely used for the analysis of trace analytes in macro and meso samples. Major and minor analytes can be determined by diluting the sample before analysis, while concentrating a sample may allow for the analysis of ultratrace analytes. The scale of operations for infrared absorption is generally poorer than that for UV/Vis absorption.

Note

See Figure 3.5 to review the meaning of macro and meso for describing samples, and the meaning of major, minor, and ultratrace for describing analytes.

Accuracy

Under normal conditions a relative error of 1–5% is easy to obtained with UV/Vis absorption. Accuracy is usually limited by the quality of the blank. Examples of the type of problems that may be encountered include the presence of particulates in a sample that scatter radiation and interferents that react with analytical reagents. In the latter case the interferent may react to form an absorbing species, giving rise to a positive determinate error. Interferents also may prevent the analyte from reacting, leading to a negative determinate error. With care, it may be possible to improve the accuracy of an analysis by as much as an order of magnitude.

Precision

In absorption spectroscopy, precision is limited by indeterminate errors—primarily instrumental noise—introduced when measuring absorbance. Precision is generally worse for low absorbances where P₀ ≈ P_T, and for high absorbances when P_T approaches 0. We might expect, therefore, that precision will vary with transmittance.

We can derive an expression between precision and transmittance by applying the propagation of uncertainty as described in Chapter 4. To do so we rewrite Beer’s law asC=−1εblogT(10.22)(10.22)C=−1εblog⁡T

Table 4.10 in Chapter 4 helps us in completing the propagation of uncertainty for equation 10.22, giving the absolute uncertainty in the concentration, s_C, assC=−0.4343εb×sTT(10.23)(10.23)sC=−0.4343εb×sTT

where s_T is the absolute uncertainty in the transmittance. Dividing equation 10.23 by equation 10.22 gives the relative uncertainty in concentration, s_C/C, assCC=0.4343sTTlogT(4.8.36)(4.8.36)sCC=0.4343sTTlog⁡T

If we know the absolute uncertainty in transmittance, we can determine the relative uncertainty in concentration for any transmittance.

Determining the relative uncertainty in concentration is complicated because s_T may be a function of the transmittance. As shown in Table 10.8, three categories of indeterminate instrumental error have been observed.¹² A constant s_T is observed for the uncertainty associated with reading %T on a meter’s analog or digital scale. Typical values are ±0.2–0.3% (a k₁ of ±0.002–0.003) for an analog scale, and ±0.001% a (k₁ of ±0.000 01) for a digital scale. A constant s_T also is observed for the thermal transducers used in infrared spectrophotometers. The effect of a constant s_T on the relative uncertainty in concentration is shown by curve A in Figure 10.40. Note that the relative uncertainty is very large for both high and low absorbances, reaching a minimum when the absorbance is 0.4343. This source of indeterminate error is important for infrared spectrophotometers and for inexpensive UV/Vis spectrophotometers. To obtain a relative uncertainty in concentration of ±1–2%, the absorbance must be kept within the range 0.1–1.

Values of s_T are a complex function of transmittance when indeterminate errors are dominated by the noise associated with photon detectors. Curve B in Figure 10.40 shows that the relative uncertainty in concentration is very large for low absorbances, but is less at higher absorbances. Although the relative uncertainty reaches a minimum when the absorbance is 0.963, there is little change in the relative uncertainty for absorbances within the range 0.5–2. This source of indeterminate error generally limits the precision of high quality UV/Vis spectrophotometers for mid-to-high absorbances.

Figure 10.40 Percent relative uncertainty in concentration as a function of absorbance for the categories of indeterminate errors in Table 10.8. A: k₁ = ±0.0030; B: k₂ = ±0.0030; C:k₃ = ±0.0130. The dashed lines correspond to the minimum uncertainty for curve A (absorbance of 0.4343) and for curve B (absorbance of 0.963).

Finally, the value of s_T is directly proportional to transmittance for indeterminate errors resulting from fluctuations in the source’s intensity and from uncertainty in positioning the sample within the spectrometer. The latter is particularly important because the optical properties of any sample cell are not uniform. As a result, repositioning the sample cell may lead to a change in the intensity of transmitted radiation. As shown by curve C in Figure 10.40, the effect is only important at low absorbances. This source of indeterminate errors is usually the limiting factor for high quality UV/Vis spectrophotometers when the absorbance is relatively small.

When the relative uncertainty in concentration is limited by the %T readout resolution, the precision of the analysis can be improved by redefining 100% T and 0% T. Normally 100% T is established using a blank and 0% T is established while preventing the source’s radiation from reaching the detector. If the absorbance is too high, precision can be improved by resetting 100% T using a standard solution of the analyte whose concentration is less than that of the sample (Figure 10.41a). For a sample whose absorbance is too low, precision can be improved by redefining 0% T using a standard solution of the analyte whose concentration is greater than that of the analyte (Figure 10.41b). In this case a calibration curve is required because a linear relationship between absorbance and concentration no longer exists. Precision can be further increased by combining these two methods (Figure 10.41c). Again, a calibration curve is necessary since the relationship between absorbance and concentration is no longer linear.

Figure 10.41 Methods for improving the precision of absorption methods: (a) high-absorbance method; (b) low-absorbance method; (c) maximum precision method.

Sensitivity

The sensitivity of a molecular absorption method, which is the slope of a Beer’s law calibration curve, is the product of the analyte’s absorptivity and the pathlength of the sample cell (εb). You can improve a method’s sensitivity by selecting a wavelength where absorbance is at a maximum or by increasing the pathlength.

Note

See Figure 10.24 for an example of how the choice of wavelength affects a calibration curve’s sensitivity.

Selectivity

Selectivity is rarely a problem in molecular absorption spectrophotometry. In many cases it is possible to find a wavelength where only the analyte absorbs. When two or more species do contribute to the measured absorbance, a multicomponent analysis is still possible, as shown in Example 10.6 and Example 10.7.

Time, Cost, and Equipment

The analysis of a sample by molecular absorption spectroscopy is relatively rapid, although additional time may be required if we need to chemically convert a nonabsorbing analyte into an absorbing form. The cost of UV/Vis instrumentation ranges from several hundred dollars for a simple filter photometer, to more than $50,000 for a computer controlled high resolution, double-beam instrument equipped with variable slits, and operating over an extended range of wavelengths. Fourier transform infrared spectrometers can be obtained for as little as $15,000–$20,000, although more expensive models are available.

Contributors

David Harvey (DePauw University)

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Atomic force microscopy (AFM) or scanning force microscopy (SFM) is a very-high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction-limit.

Atomic force microscopy is a type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit. The information is gathered by “feeling” or “touching” the surface with a mechanical probe. Piezoelectric elements that facilitate tiny but accurate and precise movements on (electronic) command enable precise scanning.

Abilities

Atomic Force Microscope

The AFM has three major abilities: force measurement, topographic imaging, and manipulation.

In force measurement, AFMs can be used to measure the forces between the probe and the sample as a function of their mutual separation. This can be applied to perform force spectroscopy, to measure the mechanical properties of the sample, such as the sample’s Young’s modulus, a measure of stiffness.

For imaging, the reaction of the probe to the forces that the sample imposes on it can be used to form an image of the three-dimensional shape (topography) of a sample surface at a high resolution. This is achieved by raster scanning the position of the sample with respect to the tip and recording the height of the probe that corresponds to a constant probe-sample interaction (see section topographic imaging in AFM for more details). The surface topography is commonly displayed as a pseudocolor plot. Although the initial publication about the atomic force microscopy by Binnig, Quate and Gerber in 1986 speculated about the possibility of achieving atomic resolution, profound experimental challenges needed to be overcome before atomic resolution of defects and step edges in ambient (liquid) conditions was demonstrated in 1993 by Ohnesorge and Binnig. True atomic resolution of the silicon 7×7 surface – the atomic images of this surface obtained by STM had convinced the scientific community of the spectacular spatial resolution of scanning tunneling microscopy – had to wait a little longer before it was shown by Giessibl,

In manipulation, the forces between tip and sample can also be used to change the properties of the sample in a controlled way. Examples of this include atomic manipulation, scanning probe lithography and local stimulation of cells.

Simultaneous with the acquisition of topographical images, other properties of the sample can be measured locally and displayed as an image, often with similarly high resolution. Examples of such properties are mechanical properties like stiffness or adhesion strength and electrical properties such as conductivity or surface potential. In fact, the majority of SPM techniques are extensions of AFM that use this modality.

Other microscopy technologies

The major difference between atomic force microscopy and competing technologies such as optical microscopy and electron microscopy is that AFM does not use lenses or beam irradiation. Therefore, it does not suffer from a limitation in spatial resolution due to diffraction and aberration, and preparing a space for guiding the beam (by creating a vacuum) and staining the sample are not necessary.

There are several types of scanning microscopy including scanning probe microscopy (which includes AFM, scanning tunneling microscopy (STM) and near-field scanning optical microscope (SNOM/NSOM), STED microscopy (STED), and scanning electron microscopy and electrochemical AFM, EC-AFM). Although SNOM and STED use visible, infrared or even terahertz light to illuminate the sample, their resolution is not constrained by the diffraction limit.

Configuration

Fig. 3 shows an AFM, which typically consists of the following features. Numbers in parentheses correspond to numbered features in Fig. 3. Coordinate directions are defined by the coordinate system (0).Fig. 3: Typical configuration of an AFM.
(1): Cantilever, (2): Support for cantilever, (3): Piezoelectric element (to oscillate cantilever at its eigen frequency), (4): Tip (Fixed to open end of a cantilever, acts as the probe), (5): Detector of deflection and motion of the cantilever, (6): Sample to be measured by AFM, (7): xyz drive, (moves sample (6) and stage (8) in x, y, and z directions with respect to a tip apex (4)), and (8): Stage.

The small spring-like cantilever (1) is carried by the support (2). Optionally, a piezoelectric element (typically made of a ceramic material) (3) oscillates the cantilever (1). The sharp tip (4) is fixed to the free end of the cantilever (1). The detector (5) records the deflection and motion of the cantilever (1). The sample (6) is mounted on the sample stage (8). An xyz drive (7) permits to displace the sample (6) and the sample stage (8) in x, y, and z directions with respect to the tip apex (4). Although Fig. 3 shows the drive attached to the sample, the drive can also be attached to the tip, or independent drives can be attached to both, since it is the relative displacement of the sample and tip that needs to be controlled. Controllers and plotter are not shown in Fig. 3.

According to the configuration described above, the interaction between tip and sample, which can be an atomic scale phenomenon, is transduced into changes of the motion of cantilever which is a macro scale phenomenon. Several different aspects of the cantilever motion can be used to quantify the interaction between the tip and sample, most commonly the value of the deflection, the amplitude of an imposed oscillation of the cantilever, or the shift in resonance frequency of the cantilever (see section Imaging Modes).

Detector

The detector (5) of AFM measures the deflection (displacement with respect to the equilibrium position) of the cantilever and converts it into an electrical signal. The intensity of this signal will be proportional to the displacement of the cantilever.

Various methods of detection can be used, e.g. interferometry, optical levers, the piezoresistive method, the piezoelectric method, and STM-based detectors (see section “AFM cantilever deflection measurement”.).

Image formation

Note: The following paragraphs assume that ‘contact mode’ is used (see section Imaging Modes). For other imaging modes, the process is similar, except that ‘deflection’ should be replaced by the appropriate feedback variable.

When using the AFM to image a sample, the tip is brought into contact with the sample, and the sample is raster scanned along an x–y grid (fig 4). Most commonly, an electronic feedback loop is employed to keep the probe-sample force constant during scanning. This feedback loop has the cantilever deflection as input, and its output controls the distance along the z axis between the probe support (2 in fig. 3) and the sample support (8 in fig 3). As long as the tip remains in contact with the sample, and the sample is scanned in the x–y plane, height variations in the sample will change the deflection of the cantilever. The feedback then adjusts the height of the probe support so that the deflection is restored to a user-defined value (the setpoint). A properly adjusted feedback loop adjusts the support-sample separation continuously during the scanning motion, such that the deflection remains approximately constant. In this situation, the feedback output equals the sample surface topography to within a small error.

Historically, a different operation method has been used, in which the sample-probe support distance is kept constant and not controlled by a feedback (servo mechanism). In this mode, usually referred to as ‘constant height mode’, the deflection of the cantilever is recorded as a function of the sample x–y position. As long as the tip is in contact with the sample, the deflection then corresponds to surface topography. The main reason this method is not very popular anymore, is that the forces between tip and sample are not controlled, which can lead to forces high enough to damage the tip or the sample. It is however common practice to record the deflection even when scanning in ‘constant force mode’, with feedback. This reveals the small tracking error of the feedback, and can sometimes reveal features that the feedback was not able to adjust for.

The AFM signals, such as sample height or cantilever deflection, are recorded on a computer during the x–y scan. They are plotted in a pseudocolor image, in which each pixel represents an x–y position on the sample, and the color represents the recorded signal.Fig. 5: Topographic image forming by AFM.
(1): Tip apex, (2): Sample surface, (3): Z-orbit of Tip apex, (4): Cantilever.

History

The AFM was invented by IBM scientists in 1985, The precursor to the AFM, the scanning tunneling microscope (STM), was developed by Gerd Binnig and Heinrich Rohrer in the early 1980s at IBM Research – Zurich, a development that earned them the 1986 Nobel Prize for Physics. Binnig invented the atomic force microscope and the first experimental implementation was made by Binnig, Quate and Gerber in 1986.

The first commercially available atomic force microscope was introduced in 1989. The AFM is one of the foremost tools for imaging, measuring, and manipulating matter at the nanoscale.

Applications

The AFM has been applied to problems in a wide range of disciplines of the natural sciences, including solid-state physics, semiconductor science and technology, molecular engineering, polymer chemistry and physics, surface chemistry, molecular biology, cell biology, and medicine.

Applications in the field of solid state physics include (a) the identification of atoms at a surface, (b) the evaluation of interactions between a specific atom and its neighboring atoms, and (c) the study of changes in physical properties arising from changes in an atomic arrangement through atomic manipulation.

In molecular biology, AFM can be used to study the structure and mechanical properties of protein complexes and assemblies. For example, AFM has been used to image microtubules and measure their stiffness.

In cellular biology, AFM can be used to attempt to distinguish cancer cells and normal cells based on a hardness of cells, and to evaluate interactions between a specific cell and its neighboring cells in a competitive culture system. AFM can also be used to indent cells, to study how they regulate the stiffness or shape of the cell membrane or wall.

In some variations, electric potentials can also be scanned using conducting cantilevers. In more advanced versions, currents can be passed through the tip to probe the electrical conductivity or transport of the underlying surface, but this is a challenging task with few research groups reporting consistent data (as of 2004).

Principles

Electron micrograph of a used AFM cantilever. Image width ~100 micrometers

Electron micrograph of a used AFM cantilever. Image width ~30 micrometers

The AFM consists of a cantilever with a sharp tip (probe) at its end that is used to scan the specimen surface. The cantilever is typically silicon or silicon nitridewith a tip radius of curvature on the order of nanometers. When the tip is brought into proximity of a sample surface, forces between the tip and the sample lead to a deflection of the cantilever according to Hooke’s law. Depending on the situation, forces that are measured in AFM include mechanical contact force, van der Waals forces, capillary forces, chemical bonding, electrostatic forces, magnetic forces (see magnetic force microscope, MFM), Casimir forces, solvation forces, etc. Along with force, additional quantities may simultaneously be measured through the use of specialized types of probes. Atomic force microscope topographical scan of a glass surface. The micro and nano-scale features of the glass can be observed, portraying the roughness of the material. The image space is (x,y,z) = (20 µm × 20 µm × 420 nm).

The AFM can be operated in a number of modes, depending on the application. In general, possible imaging modes are divided into static (also called contact) modes and a variety of dynamic (non-contact or “tapping”) modes where the cantilever is vibrated or oscillated at a given frequency.^[7]

Imaging modes

AFM operation is usually described as one of three modes, according to the nature of the tip motion: contact mode, also called static mode (as opposed to the other two modes, which are called dynamic modes); tapping mode, also called intermittent contact, AC mode, or vibrating mode, or, after the detection mechanism, amplitude modulation AFM; non-contact mode, or, again after the detection mechanism, frequency modulation AFM.

Despite the nomenclature, repulsive contact can occur or be avoided both in amplitude modulation AFM and frequency modulation AFM, depending on the settings.

Contact mode

In contact mode, the tip is “dragged” across the surface of the sample and the contours of the surface are measured either using the deflection of the cantilever directly or, more commonly, using the feedback signal required to keep the cantilever at a constant position. Because the measurement of a static signal is prone to noise and drift, low stiffness cantilevers (i.e. cantilevers with a low spring constant, k) are used to achieve a large enough deflection signal while keeping the interaction force low. Close to the surface of the sample, attractive forces can be quite strong, causing the tip to “snap-in” to the surface. Thus, contact mode AFM is almost always done at a depth where the overall force is repulsive, that is, in firm “contact” with the solid surface.

Tapping mode

Single polymer chains (0.4 nm thick) recorded in a tapping mode under aqueous media with different pH.

In ambient conditions, most samples develop a liquid meniscus layer. Because of this, keeping the probe tip close enough to the sample for short-range forces to become detectable while preventing the tip from sticking to the surface presents a major problem for contact mode in ambient conditions. Dynamic contact mode (also called intermittent contact, AC mode or tapping mode) was developed to bypass this problem. Nowadays, tapping mode is the most frequently used AFM mode when operating in ambient conditions or in liquids.

In tapping mode, the cantilever is driven to oscillate up and down at or near its resonance frequency. This oscillation is commonly achieved with a small piezo element in the cantilever holder, but other possibilities include an AC magnetic field (with magnetic cantilevers), piezoelectric cantilevers, or periodic heating with a modulated laser beam. The amplitude of this oscillation usually varies from several nm to 200 nm. In tapping mode, the frequency and amplitude of the driving signal are kept constant, leading to a constant amplitude of the cantilever oscillation as long as there is no drift or interaction with the surface. The interaction of forces acting on the cantilever when the tip comes close to the surface, Van der Waals forces, dipole-dipole interactions, electrostatic forces, etc. cause the amplitude of the cantilever’s oscillation to change (usually decrease) as the tip gets closer to the sample. This amplitude is used as the parameter that goes into the electronic servo that controls the height of the cantilever above the sample. The servo adjusts the height to maintain a set cantilever oscillation amplitude as the cantilever is scanned over the sample. A tapping AFM image is therefore produced by imaging the force of the intermittent contacts of the tip with the sample surface.

Although the peak forces applied during the contacting part of the oscillation can be much higher than typically used in contact mode, tapping mode generally lessens the damage done to the surface and the tip compared to the amount done in contact mode. This can be explained by the short duration of the applied force, and because the lateral forces between tip and sample are significantly lower in tapping mode over contact mode. Tapping mode imaging is gentle enough even for the visualization of supported lipid bilayers or adsorbed single polymer molecules (for instance, 0.4 nm thick chains of synthetic polyelectrolytes) under liquid medium. With proper scanning parameters, the conformation of single molecules can remain unchanged for hours, and even single molecular motors can be imaged while moving.

When operating in tapping mode, the phase of the cantilever’s oscillation with respect to the driving signal can be recorded as well. This signal channel contains information about the energy dissipated by the cantilever in each oscillation cycle. Samples that contain regions of varying stiffness or with different adhesion properties can give a contrast in this channel that is not visible in the topographic image. Extracting the sample’s material properties in a quantitative manner from phase images, however, is often not feasible.

Non-contact mode

In non-contact atomic force microscopy mode, the tip of the cantilever does not contact the sample surface. The cantilever is instead oscillated at either its resonant frequency(frequency modulation) or just above (amplitude modulation) where the amplitude of oscillation is typically a few nanometers (<10 nm) down to a few picometers. The van der Waals forces, which are strongest from 1 nm to 10 nm above the surface, or any other long-range force that extends above the surface acts to decrease the resonance frequency of the cantilever. This decrease in resonant frequency combined with the feedback loop system maintains a constant oscillation amplitude or frequency by adjusting the average tip-to-sample distance. Measuring the tip-to-sample distance at each (x,y) data point allows the scanning software to construct a topographic image of the sample surface.

Non-contact mode AFM does not suffer from tip or sample degradation effects that are sometimes observed after taking numerous scans with contact AFM. This makes non-contact AFM preferable to contact AFM for measuring soft samples, e.g. biological samples and organic thin film. In the case of rigid samples, contact and non-contact images may look the same. However, if a few monolayers of adsorbed fluid are lying on the surface of a rigid sample, the images may look quite different. An AFM operating in contact mode will penetrate the liquid layer to image the underlying surface, whereas in non-contact mode an AFM will oscillate above the adsorbed fluid layer to image both the liquid and surface.

Schemes for dynamic mode operation include frequency modulation where a phase-locked loop is used to track the cantilever’s resonance frequency and the more common amplitude modulation with a servo loop in place to keep the cantilever excitation to a defined amplitude. In frequency modulation, changes in the oscillation frequency provide information about tip-sample interactions. Frequency can be measured with very high sensitivity and thus the frequency modulation mode allows for the use of very stiff cantilevers. Stiff cantilevers provide stability very close to the surface and, as a result, this technique was the first AFM technique to provide true atomic resolution in ultra-high vacuumconditions.

In amplitude modulation, changes in the oscillation amplitude or phase provide the feedback signal for imaging. In amplitude modulation, changes in the phase of oscillation can be used to discriminate between different types of materials on the surface. Amplitude modulation can be operated either in the non-contact or in the intermittent contact regime. In dynamic contact mode, the cantilever is oscillated such that the separation distance between the cantilever tip and the sample surface is modulated.

Amplitude modulation has also been used in the non-contact regime to image with atomic resolution by using very stiff cantilevers and small amplitudes in an ultra-high vacuum environment.

Topographic image

Image formation is a plotting method that produces a color mapping through changing the x–y position of the tip while scanning and recording the measured variable, i.e. the intensity of control signal, to each x–y coordinate. The color mapping shows the measured value corresponding to each coordinate. The image expresses the intensity of a value as a hue. Usually, the correspondence between the intensity of a value and a hue is shown as a color scale in the explanatory notes accompanying the image.

What is the topographic image of atomic force microscope?

Operation mode of image forming of the AFM are generally classified into two groups from the viewpoint whether it uses z-Feedback loop (not shown) to maintain the tip-sample distance to keep signal intensity exported by the detector. The first one (using z-Feedback loop), said to be “constant XX mode” (XX is something which kept by z-Feedback loop).

Topographic image formation mode is based on abovementioned “constant XX mode”, z-Feedback loop controls the relative distance between the probe and the sample through outputting control signals to keep constant one of frequency, vibration and phase which typically corresponds to the motion of cantilever (for instance, voltage is applied to the Z-piezoelectric element and it moves the sample up and down towards the Z direction.

Details will be explained in the case that especially “constant df mode”(FM-AFM) among AFM as an instance in next section.

Topographic image of FM-AFM

When the distance between the probe and the sample is brought to the range where atomic force may be detected, while a cantilever is excited in its natural eigen frequency (f₀), a phenomenon occurs that the resonance frequency (f) of the cantilever shifts from its original resonance frequency (natural eigen frequency). In other words, in the range where atomic force may be detected, the frequency shift (df=f-f₀) will be observed. So, when the distance between the probe and the sample is in the non-contact region, the frequency shift increases in negative direction as the distance between the probe and the sample gets smaller.

When the sample has concavity and convexity, the distance between the tip-apex and the sample varies in accordance with the concavity and convexity accompanied with a scan of the sample along x–y direction (without height regulation in z-direction). As a result, the frequency shift arises. The image in which the values of the frequency obtained by a raster scan along the x–y direction of the sample surface are plotted against the x–y coordination of each measurement point is called a constant-height image.

On the other hand, the df may be kept constant by moving the probe upward and downward (See (3) of FIG.5) in z-direction using a negative feedback (by using z-feedback loop) while the raster scan of the sample surface along the x–y direction. The image in which the amounts of the negative feedback (the moving distance of the probe upward and downward in z-direction) are plotted against the x–y coordination of each measurement point is a topographic image. In other words, the topographic image is a trace of the tip of the probe regulated so that the df is constant and it may also be considered to be a plot of a constant-height surface of the df.

Therefore, the topographic image of the AFM is not the exact surface morphology itself, but actually the image influenced by the bond-order between the probe and the sample, however, the topographic image of the AFM is considered to reflect the geographical shape of the surface more than the topographic image of a scanning tunnel microscope.

Force spectroscopy

Another major application of AFM (besides imaging) is force spectroscopy, the direct measurement of tip-sample interaction forces as a function of the gap between the tip and sample (the result of this measurement is called a force-distance curve). For this method, the AFM tip is extended towards and retracted from the surface as the deflection of the cantilever is monitored as a function of piezoelectric displacement. These measurements have been used to measure nanoscale contacts, atomic bonding, Van der Waals forces, and Casimir forces, dissolution forces in liquids and single molecule stretching and rupture forces. Furthermore, AFM was used to measure, in an aqueous environment, the dispersion force due to polymer adsorbed on the substrate. Forces of the order of a few piconewtons can now be routinely measured with a vertical distance resolution of better than 0.1 nanometers. Force spectroscopy can be performed with either static or dynamic modes. In dynamic modes, information about the cantilever vibration is monitored in addition to the static deflection.

Problems with the technique include no direct measurement of the tip-sample separation and the common need for low-stiffness cantilevers, which tend to ‘snap’ to the surface. These problems are not insurmountable. An AFM that directly measures the tip-sample separation has been developed. The snap-in can be reduced by measuring in liquids or by using stiffer cantilevers, but in the latter case a more sensitive deflection sensor is needed. By applying a small dither to the tip, the stiffness (force gradient) of the bond can be measured as well.

Biological applications and other

Force spectroscopy is used in biophysics to measure the mechanical properties of living material (such as tissue or cells) or detect structures of different stiffness buried into the bulk of the sample using the stiffness tomography. Another application was to measure the interaction forces between from one hand a material stuck on the tip of the cantilever, and from another hand the surface of particles either free or occupied by the same material. From the adhesion force distribution curve, a mean value of the forces has been derived. It allowed to make a cartography of the surface of the particles, covered or not by the material. AFM have been also used for mechanically unfolding proteins. In such experiments, the analyzes of the mean unfolding forces with the appropriate model leads to the obtainment of the information about the unfolding rate and free energy profile parameters of the protein.

Identification of individual surface atoms

The AFM can be used to image and manipulate atoms and structures on a variety of surfaces. The atom at the apex of the tip “senses” individual atoms on the underlying surface when it forms incipient chemical bonds with each atom. Because these chemical interactions subtly alter the tip’s vibration frequency, they can be detected and mapped. This principle was used to distinguish between atoms of silicon, tin and lead on an alloy surface, by comparing these ‘atomic fingerprints’ to values obtained from large-scale density functional theory (DFT) simulations.

The trick is to first measure these forces precisely for each type of atom expected in the sample, and then to compare with forces given by DFT simulations. The team found that the tip interacted most strongly with silicon atoms, and interacted 24% and 41% less strongly with tin and lead atoms, respectively. Thus, each different type of atom can be identified in the matrix as the tip is moved across the surface.

Probe

An AFM probe has a sharp tip on the free-swinging end of a cantilever that is protruding from a holder. The dimensions of the cantilever are in the scale of micrometers. The radius of the tip is usually on the scale of a few nanometers to a few tens of nanometers. (Specialized probes exist with much larger end radii, for example probes for indentation of soft materials.) The cantilever holder, also called holder chip – often 1.6 mm by 3.4 mm in size – allows the operator to hold the AFM cantilever/probe assembly with tweezers and fit it into the corresponding holder clips on the scanning head of the atomic force microscope.

This device is most commonly called an “AFM probe”, but other names include “AFM tip” and “cantilever” (employing the name of a single part as the name of the whole device). An AFM probe is a particular type of SPM (scanning probe microscopy) probe.

AFM probes are manufactured with MEMS technology. Most AFM probes used are made from silicon (Si), but borosilicate glass and silicon nitride are also in use. AFM probes are considered consumables as they are often replaced when the tip apex becomes dull or contaminated or when the cantilever is broken. They can cost from a couple of tens of dollars up to hundreds of dollars per cantilever for the most specialized cantilever/probe combinations.

Just the tip is brought very close to the surface of the object under investigation, the cantilever is deflected by the interaction between the tip and the surface, which is what the AFM is designed to measure. A spatial map of the interaction can be made by measuring the deflection at many points on a 2D surface.

Several types of interaction can be detected. Depending on the interaction under investigation, the surface of the tip of the AFM probe needs to be modified with a coating. Among the coatings used are gold – for covalent bonding of biological molecules and the detection of their interaction with a surface, diamond for increased wear resistance and magnetic coatings for detecting the magnetic properties of the investigated surface. Another solution exists to achieve high resolution magnetic imaging : having the probe equip with a microSQUID. The AFM tips is fabricated using silicon micro machining and the precise positioning of the microSQUID loop is done by electron beam lithography.

The surface of the cantilevers can also be modified. These coatings are mostly applied in order to increase the reflectance of the cantilever and to improve the deflection signal.

Forces vs tip geometry

The forces between the tip and the sample strongly depend on the geometry of the tip. Various studies were exploited in the past years to write the forces as a function of the tip parameters.

Among the different forces between the tip and the sample, the water meniscus forces are highly interesting, both in air and liquid environment. Other forces must be considered, like the Coulomb force, van der Waals forces, double layer interactions, solvation forces, hydration and hydrophobic forces.

Water meniscus

Water meniscus forces are highly interesting for AFM measurements in air. Due to the ambient humidity, a thin layer of water is formed between the tip and the sample during air measurements. The resulting capillary force gives rise to a strong attractive force that pulls the tip onto the surface. In fact, the adhesion force measured between tip and sample in ambient air of finite humidity is usually dominated by capillary forces. As a consequence, it is difficult to pull the tip away from the surface. For soft samples including many polymers and in particular biological materials, the strong adhesive capillary force gives rise to sample degradation and destruction upon imaging in contact mode. Historically, these problems were an important motivation for the development of dynamic imaging in air (e.g. ‘tapping mode’). During tapping mode imaging in air, capillary bridges still form. Yet, for suitable imaging conditions, the capillary bridges are formed and broken in every oscillation cycle of the cantilever normal to the surface, as can be inferred from an analysis of cantilever amplitude and phase vs. distance curves. As a consequence, destructive shear forces are largely reduced and soft samples can be investigated.

In order to quantify the equilibrium capillary force, it is necessary to start from the Laplace equation for pressure:Model for AFM water meniscus

{\displaystyle P=\gamma _{L}({\frac {1}{r}}_{1}+{\frac {1}{r}}_{0})\simeq {\frac {\gamma _{L}}{r_{eff}}}}

where γ_L is the surface energy and r₀ and r₁ are defined in the figure.

The pressure is applied on an area of

{\displaystyle A\simeq 2\pi R\simeq [r_{eff}(1+\cos \theta )+h]}

where d, θ, and h are defined in the figure.

The force which pulles together the two surfaces is

{\displaystyle F=2\pi R\gamma _{L}(1+\cos \theta +{\frac {h}{r_{eff}}})}

The same formula could also be calculated as a function of relative humidity.

Gao calculated formulas for different tip geometries. As an example, the forse decreases by 20% for a conical tip with respect to a spherical tip.

When these forces are calculated, a difference must be made between the wet on dry situation and the wet on wet situation.

For a spherical tip, the force is:

{\displaystyle f_{m}=-2\pi R\gamma _{L}(\cos \theta +\cos \phi )(1-{\frac {dh}{dD}})} for dry on wet

{\displaystyle f_{m}=-2\pi R\gamma _{L}{\frac {dr_{0}}{dD}}}for wet on wet

where θ is the contact angle of the dry sphere and φ is the immersed angle, as shown in the figure Also R,h and D are illustrated in the same figure.

For a conical tip, the formula becomes:

{\displaystyle f_{m}=-2\pi R\gamma _{L}{\frac {\tan \delta }{\cos \delta }}(\cos \theta +\sin \delta )(hD)(1-{\frac {dh}{dD}})} for dry on wet

{\displaystyle f_{m}=-2\pi R\gamma _{L}({\frac {1}{\cos \delta }}+\sin \delta )(r_{0})({\frac {dr_{0}}{dD}})} for wet on wet

where δ is the half cone angle and r₀ and h are parameters of the meniscus profile.

AFM cantilever-deflection measurement

Beam-deflection measurement

AFM beam-deflection detection

The most common method for cantilever-deflection measurements is the beam-deflection method. In this method, laser light from a solid-state diode is reflected off the back of the cantilever and collected by a position-sensitive detector (PSD) consisting of two closely spaced photodiodes, whose output signal is collected by a differential amplifier. Angular displacement of the cantilever results in one photodiode collecting more light than the other photodiode, producing an output signal (the difference between the photodiode signals normalized by their sum), which is proportional to the deflection of the cantilever. The sensitivity of the beam-deflection method is very high, a noise floor on the order of 10 fm Hz^{−​1⁄₂} can be obtained routinely in a well-designed system. Although this method is sometimes called the ‘optical lever’ method, the signal is not amplified if the beam path is made longer. A longer beam path increases the motion of the reflected spot on the photodiodes, but also widens the spot by the same amount due to diffraction, so that the same amount of optical power is moved from one photodiode to the other. The ‘optical leverage’ (output signal of the detector divided by deflection of the cantilever) is inversely proportional to the numerical aperture of the beam focusing optics, as long as the focused laser spot is small enough to fall completely on the cantilever. It is also inversely proportional to the length of the cantilever.

The relative popularity of the beam-deflection method can be explained by its high sensitivity and simple operation, and by the fact that cantilevers do not require electrical contacts or other special treatments, and can therefore be fabricated relatively cheaply with sharp integrated tips.

Other deflection-measurement methods

Many other methods for beam-deflection measurements exist.

• Piezoelectric detection – Cantilevers made from quartz (such as the qPlus configuration), or other piezoelectric materials can directly detect deflection as an electrical signal. Cantilever oscillations down to 10pm have been detected with this method.
• Laser Doppler vibrometry – A laser Doppler vibrometer can be used to produce very accurate deflection measurements for an oscillating cantilever (thus is only used in non-contact mode). This method is expensive and is only used by relatively few groups.
• Scanning tunneling microscope (STM) — The first atomic microscope used an STM complete with its own feedback mechanism to measure deflection. This method is very difficult to implement, and is slow to react to deflection changes compared to modern methods.
• Optical interferometry – Optical interferometry can be used to measure cantilever deflection. Due to the nanometre scale deflections measured in AFM, the interferometer is running in the sub-fringe regime, thus, any drift in laser power or wavelength has strong effects on the measurement. For these reasons optical interferometer measurements must be done with great care (for example using index matching fluids between optical fibre junctions), with very stable lasers. For these reasons optical interferometry is rarely used.
• Capacitive detection – Metal coated cantilevers can form a capacitor with another contact located behind the cantilever. Deflection changes the distance between the contacts and can be measured as a change in capacitance.
• Piezoresistive detection – Cantilevers can be fabricated with piezoresistive elements that act as a strain gauge. Using a Wheatstone bridge, strain in the AFM cantilever due to deflection can be measured. This is not commonly used in vacuum applications, as the piezoresistive detection dissipates energy from the system affecting Q of the resonance.

Piezoelectric scanners

AFM scanners are made from piezoelectric material, which expands and contracts proportionally to an applied voltage. Whether they elongate or contract depends upon the polarity of the voltage applied. Traditionally the tip or sample is mounted on a ‘tripod’ of three piezo crystals, with each responsible for scanning in the x,y and z directions. In 1986, the same year as the AFM was invented, a new piezoelectric scanner, the tube scanner, was developed for use in STM. Later tube scanners were incorporated into AFMs. The tube scanner can move the sample in the x, y, and z directions using a single tube piezo with a single interior contact and four external contacts. An advantage of the tube scanner compared to the original tripod design, is better vibrational isolation, resulting from the higher resonant frequency of the single element construction, in combination with a low resonant frequency isolation stage. A disadvantage is that the x–y motion can cause unwanted z motion resulting in distortion. Another popular design for AFM scanners is the flexurestage, which uses separate piezos for each axis, and couples them through a flexure mechanism.

Scanners are characterized by their sensitivity, which is the ratio of piezo movement to piezo voltage, i.e., by how much the piezo material extends or contracts per applied volt. Because of differences in material or size, the sensitivity varies from scanner to scanner. Sensitivity varies non-linearly with respect to scan size. Piezo scanners exhibit more sensitivity at the end than at the beginning of a scan. This causes the forward and reverse scans to behave differently and display hysteresis between the two scan directions.This can be corrected by applying a non-linear voltage to the piezo electrodes to cause linear scanner movement and calibrating the scanner accordingly. One disadvantage of this approach is that it requires re-calibration because the precise non-linear voltage needed to correct non-linear movement will change as the piezo ages (see below). This problem can be circumvented by adding a linear sensor to the sample stage or piezo stage to detect the true movement of the piezo. Deviations from ideal movement can be detected by the sensor and corrections applied to the piezo drive signal to correct for non-linear piezo movement. This design is known as a ‘closed loop’ AFM. Non-sensored piezo AFMs are referred to as ‘open loop’ AFMs.

The sensitivity of piezoelectric materials decreases exponentially with time. This causes most of the change in sensitivity to occur in the initial stages of the scanner’s life. Piezoelectric scanners are run for approximately 48 hours before they are shipped from the factory so that they are past the point where they may have large changes in sensitivity. As the scanner ages, the sensitivity will change less with time and the scanner would seldom require recalibration, though various manufacturer manuals recommend monthly to semi-monthly calibration of open loop AFMs.

Advantages and disadvantages

The first atomic force microscope

Advantages

AFM has several advantages over the scanning electron microscope (SEM). Unlike the electron microscope, which provides a two-dimensional projection or a two-dimensional image of a sample, the AFM provides a three-dimensional surface profile. In addition, samples viewed by AFM do not require any special treatments (such as metal/carbon coatings) that would irreversibly change or damage the sample, and does not typically suffer from charging artifacts in the final image. While an electron microscope needs an expensive vacuum environment for proper operation, most AFM modes can work perfectly well in ambient air or even a liquid environment. This makes it possible to study biological macromolecules and even living organisms. In principle, AFM can provide higher resolution than SEM. It has been shown to give true atomic resolution in ultra-high vacuum (UHV) and, more recently, in liquid environments. High resolution AFM is comparable in resolution to scanning tunneling microscopy and transmission electron microscopy. AFM can also be combined with a variety of optical microscopy and spectroscopy techniques such as fluorescent microscopy of infrared spectroscopy, giving rise to scanning near-field optical microscopy, nano-FTIR and further expanding its applicability. Combined AFM-optical instruments have been applied primarily in the biological sciences but have recently attracted strong interest in photovoltaics and energy-storage research, polymer sciences, nanotechnology and even medical research.

Disadvantages

A disadvantage of AFM compared with the scanning electron microscope (SEM) is the single scan image size. In one pass, the SEM can image an area on the order of square millimeters with a depth of field on the order of millimeters, whereas the AFM can only image a maximum scanning area of about 150×150 micrometers and a maximum height on the order of 10–20 micrometers. One method of improving the scanned area size for AFM is by using parallel probes in a fashion similar to that of millipede data storage.

The scanning speed of an AFM is also a limitation. Traditionally, an AFM cannot scan images as fast as an SEM, requiring several minutes for a typical scan, while an SEM is capable of scanning at near real-time, although at relatively low quality. The relatively slow rate of scanning during AFM imaging often leads to thermal drift in the imagemaking the AFM less suited for measuring accurate distances between topographical features on the image. However, several fast-acting designs were suggested to increase microscope scanning productivity including what is being termed videoAFM (reasonable quality images are being obtained with videoAFM at video rate: faster than the average SEM). To eliminate image distortions induced by thermal drift, several methods have been introduced.Showing an AFM artifact arising from a tip with a high radius of curvature with respect to the feature that is to be visualizedAFM artifact, steep sample topography

AFM images can also be affected by nonlinearity, hysteresis, and creep of the piezoelectric material and cross-talk between the x, y, zaxes that may require software enhancement and filtering. Such filtering could “flatten” out real topographical features. However, newer AFMs utilize real-time correction software (for example, feature-oriented scanning) or closed-loop scanners, which practically eliminate these problems. Some AFMs also use separated orthogonal scanners (as opposed to a single tube), which also serve to eliminate part of the cross-talk problems.

As with any other imaging technique, there is the possibility of image artifacts, which could be induced by an unsuitable tip, a poor operating environment, or even by the sample itself, as depicted on the right. These image artifacts are unavoidable; however, their occurrence and effect on results can be reduced through various methods. Artifacts resulting from a too-coarse tip can be caused for example by inappropriate handling or de facto collisions with the sample by either scanning too fast or having an unreasonably rough surface, causing actual wearing of the tip.

Due to the nature of AFM probes, they cannot normally measure steep walls or overhangs. Specially made cantilevers and AFMs can be used to modulate the probe sideways as well as up and down (as with dynamic contact and non-contact modes) to measure sidewalls, at the cost of more expensive cantilevers, lower lateral resolution and additional artifacts.

Other applications in various fields of study

AFM image of part of a Golgi apparatus isolated from HeLa cells

The latest efforts in integrating nanotechnology and biological research have been successful and show much promise for the future. Since nanoparticles are a potential vehicle of drug delivery, the biological responses of cells to these nanoparticles are continuously being explored to optimize their efficacy and how their design could be improved. Pyrgiotakis et al. were able to study the interaction between CeO₂ and Fe₂O₃ engineered nanoparticles and cells by attaching the engineered nanoparticles to the AFM tip. Studies have taken advantage of AFM to obtain further information on the behavior of live cells in biological media. Real-time atomic force spectroscopy (or nanoscopy) and dynamic atomic force spectroscopy have been used to study live cells and membrane proteins and their dynamic behavior at high resolution, on the nanoscale. Imaging and obtaining information on the topography and the properties of the cells has also given insight into chemical processes and mechanisms that occur through cell-cell interaction and interactions with other signaling molecules (ex. ligands). Evans and Calderwood used single cell force microscopy to study cell adhesion forces, bond kinetics/dynamic bond strength and its role in chemical processes such as cell signaling. Scheuring, Lévy, and Rigaud reviewed studies in which AFM to explore the crystal structure of membrane proteins of photosynthetic bacteria. Alsteen et al. have used AFM-based nanoscopy to perform a real-time analysis of the interaction between live mycobacteria and antimycobacterial drugs (specifically isoniazid, ethionamide, ethambutol, and streptomycine),^[58]which serves as an example of the more in-depth analysis of pathogen-drug interactions that can be done through AFM.

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