7.3.1 Noise sources in atomic force microscopy

The limitations of the metrological capabilities of an AFM due to thermal noise are well documented [11]. However, not only thermal but all noise sources need to be systematically investigated and their particular contributions to the total amount of the noise quantified for metrological purposes [12]. Note that most of the discussions on noise in AFM are also of relevance to other forms of SPM.

Noise sources can be either external, including:

or internal (intrinsic), including:

In practice, state-of-the-art AFMs in metrology laboratories have a resolution limited only by the temperature of the measurement. There are unfortunately no general calibration artefacts able to test this level of resolution for all AFM methods. Force and lateral resolution, as well as the overall accuracy of measurement, should be approached with caution in the absence of a clearly identified method to measure them.

It is well known that adjustments made by the user (e.g. the control loop parameters, scan field size and speed) also have a substantial influence on the measurement [13].

To reduce the total noise, the sub-components of noise must be investigated. The total amount of the z-axis noise can be determined by static or dynamic measurements [14] as described in the following section.

7.3.1.1 Static noise determination

To determine the static noise of an SPM, the probe is placed in contact with the sample, the distance is actively controlled, but the xy scan is disabled, that is the scan size is zero. The z-axis signal is recorded and analysed (e.g. root mean square (RMS) determination or calculation of the fast Fourier transform to identify dominant frequencies, which then serve to identify causes of noise). An example of a noise signal for an AFM is shown in Figure 7.3; the RMS noise is 13 p.m. in this case (represented as an Rq parameter – see Section 8.2.7.2).

7.3.2 Some common artefacts in AFM imaging

One of the reasons that AFMs only been integrated into the production environment in a few specialised applications is the presence of numerous ‘artefacts’ in their images that are not due to the surface topography of the surface being measured. Usually a high level of expertise is required to identify these artefacts. The availability of reference substrates and materials will allow industry to use AFMs (and other SPMs) more widely.

7.3.2.1 Tip size and shape

Many of the most common artefacts in AFM imaging are related to the finite size and shape of the tip. Commonly used AFM probes, such as those manufactured from silicon nitride and silicon, have pyramidal-shaped tips [15]. These tips can have a radius of curvature as small as 1 nm, but often the radius is much larger. When imaging vertical features that are several tens of nanometres or more in height, the tip half angle limits the lateral resolution. When the tip moves over a sharp feature, the sides of the tip, rather than just the tip apex, contact the edges of the feature (Figure 7.4). For features with vertical relief less than approximately 30 nm, it is the radius of curvature of the tip that limits resolution, resulting in tip broadening of the feature of interest. The resulting image is a non-linear combination of the sample shape and the tip shape. Various deconvolution (or its non-linear equivalent, erosion) methods, including commercial software packages, are available although such software must be used with caution [16–18]. There are also many physical artefacts that can be used to measure the shape of an AFM tip [19–21].

7.3.3 Determining the coordinate system of an AFM

There will always be some imperfections in the coordinate system for a given AFM. The calibration of the lateral scan axes is usually carried out using 1D or 2D lateral calibration artefacts. These artefacts are usually formed by equidistant structures with defined features whose mean spacing (the pitch) serves to calibrate the lateral axes. In Figure 7.5(a), a set of parallel regression lines along similar features of the structure is calculated. The mean distance between these lines is the pitch, p_x. In Figure 7.5(b), a set of parallel regression lines is calculated, each through a column of centres of similar features; the mean distance between these lines is the pitch, p_x in the x-direction of the grating. Similarly, another set of parallel regression lines is calculated, each through a series of centres of the grating; the mean distance of these grating lines is the pitch, p_y in the y-direction of the grating. The orthogonality of the grating is the angle formed by the p_x and p_y vectors.

Local deviations are a measure of the non-linearity of the axes. In addition, the orthogonality deviation and the crosstalk of the lateral scan axes can be determined.

For 2D lateral artefacts, it is important not to confuse the pitches, p_x and p_y, and the mean spacings, a_x and a_y, of the individual grating: p_x and a_x or p_y and a_y are identical only for perfectly orthogonal gratings. Where high-quality gratings are used, which are almost orthogonal, the difference can often be ignored in the calibration of the axes. These differences, however, become significant when a 2D artefact is used to check the orthogonality of the scanner axes.

In measurements on lateral artefacts, the selection of the scan range and the scan speed or rate is important, because the calibration factors are strongly influenced by dynamic non-linearity and image distortions [23]. This is also true for systems with active position control. In calibration, the scan speed must, therefore, be adjusted to reflect the later measurements that are to be made.

7.3.4 Traceability of atomic force microscopy

From the metrological point of view, AFMs are generally subdivided into the three categories [12]:

Another important aspect of traceability is the uncertainty of measurement (see Section 2.8.3). It is very rare to see AFM measurements quoted with an associated uncertainty as many of the points discussed in Section 6.11 apply to AFMs (and SPMs in general). Uncertainties are usually only quoted for the metrological AFMs or for simple artefacts such as step heights [28] or 1D gratings [29].

7.3.5 Force measurement with AFMs

Force measurements with an AFM are carried out by monitoring the cantilever deflection as the sample approaches, makes contact with, and then retracts from the cantilever. However, the raw cantilever deflection measurement is a measure of the deflection of the cantilever at some point and not directly of the force. For a beam deflection system, for example, the cantilever deflection is recorded in volts. An additional problem is that the distance (or separation) between the tip and the sample is not measured directly [30]; the AFM measures the displacement of the piezoelectric scanner that supports the sample. A force curve graph of cantilever deflection (in volts) and corresponding piezoelectric scanner displacement (in metres) (Figure 7.6(a)) must be interpreted to give a force–distance curve (i.e. force of interaction in units of force against separation between the sample and the cantilever in units of length (see Figure 7.6(b))). With reference to Figure 7.6(a), when the tip and sample are far apart (i) they exhibit no interaction (zero force). As the sample approaches the tip, inter-molecular forces between the tip and the sample cause the cantilever to deflect upwards (ii) due to repulsive forces (in this case between a charged substrate and tip, but attractive forces are commonly observed as well). Eventually the tip makes contact with the sample (iii) and their movement becomes coupled (region of constant compliance). The sample is then retracted from the tip (iv) until the tip/cantilever and sample return to their original positions completing one cycle. Hysteresis, shown here, may occur upon retraction due to adhesion forces. Interfacial forces are measured on approach and adhesion forces are measured upon retraction; repulsive forces are positive and attractive forces are negative.

To obtain the force part of the force–distance curve, the photodiode values are converted to force using F=k_cd, where F is the force, d is the cantilever deflection and k_c is the cantilever spring constant. To convert the cantilever deflection measured by the photodiode in volts to metres, a displacement conversion factor (also called the optical lever sensitivity) is obtained from the region of the force curve where the sample is in contact with the cantilever. For an infinitely hard contact, every displacement of the piezoelectric scanner displaces the sample or the tip; the cantilever is pushed upwards, which is recorded as a voltage output on the photodiode. The slope of the force curve in the region where the cantilever is in contact with the sample defines the optical lever sensitivity. This part of the force curve is called the region of constant compliance or region of contact.

It is important to note that using the constant compliance region of the force curve to convert photodiode response to deflection will overestimate the force of interaction if the cantilever is not the most compliant component of the system. This is often the case when soft, deformable substances such as polymers are used in force measurements (either as a sample or linked to the tip/cantilever). If a compliant substrate is used, other methods are needed to accurately convert the measured deflection of the cantilever into a force of interaction [31]. In this case, the optical lever sensitivity is determined by pressing the tip/cantilever against a hard sample (e.g. mica), before and after it is used on a soft sample. However, often this method does not work as the optical lever sensitivity is strongly dependent upon a number of factors. These factors include the position and shape of the laser spot and the difficulty in precisely aligning the laser spot on the same position on the cantilever from experiment to experiment. Also, the use of a hard sample cannot be applied if it is the tip/cantilever that supports the most compliant component of the system (e.g. a molecule attached to the cantilever). Another method that relies on the ‘photodiode shift voltage’, a parameter that is very sensitive to the position and shape of the laser of the photodetector, can be used to convert volts of cantilever deflection into metres of deflection [32]. This method ensures that forces can be determined regardless of the compliance of the cantilever relative to any other component in the AFM, and also ensures the preservation of fragile macromolecules, which may be present on the sample or attached to the cantilever.

7.3.6 AFM cantilever calibration

AFMs are sensitive to very small forces in the piconewton range. In order to measure these forces accurately, the stiffness of the probe must be determined. Stiffness calibration procedures rely on imposing known forces on the probe, measuring the geometrical and material properties of the probe, or measuring its thermal fluctuations.

The cantilever’s spring constant is essentially dependent upon its composition and dimensions [33]. Nominal values listed by manufacturers may be incorrect by an order of magnitude and it is, therefore, necessary to determine the spring constant for each cantilever or for each batch of cantilevers from a wafer [34]. Parameters such as Young’s modulus (related to composition), and cantilever length and thickness, can be used with theoretical equations to calculate a spring constant [35]. However, calculated values can be inaccurate due to the unknown material properties of the cantilever (the stoichiometry of silicon nitride, for example, can vary from Si₃N₄ to Si₅N₄ [36]). Furthermore, the measurement of cantilever thickness, which is a dominant parameter in theoretical equations, is extremely difficult. The spring constant depends on the cantilever thickness to the third power, so even small uncertainty in the thickness measurement will result in large variations in the calculated spring constant [37].

An accurate, but often destructive, way to measure spring constant is the added-mass method [38]. In this method, beads of known mass are attached to the end of the cantilever. The additional mass causes the cantilever resonant frequency to decrease proportional to the mass. A graph of added mass against resonant frequency yields a straight line with a slope corresponding to the spring constant.

A further method to determine the spring constant is the measurement of the force that an AFM imparts onto a surface by measuring the thermal fluctuations of the cantilever – in this method, the cantilever is modelled as a simple harmonic oscillator (usually only in one degree of freedom) [39]. With knowledge of the potential energy of the system and applying the equipartition theorem, the spring constant of the cantilever can be calculated from the motion of the cantilever and its surrounding heat-bath temperature. The thermal method has three major problems [40]: (i) higher vibration modes cannot be ignored, (ii) the method to measure deflection usually measures the inclination rather than the displacement and (iii) only the first modes are accessible due to the bandwidth limitations of the experiments.

For directly traceable measurements of the force an AFM cantilever imparts on a surface, electrostatic balances can be used, but they are very costly and inconvenient (see Section 10.3.3). Many of the devices discussed in Section 10.3.4 can also be used to measure spring constant when used as passive springs.

7.3.7 Inter- and intra-molecular force measurement using AFM

As discussed previously, the AFM images a sample by sensing and responding to forces between the tip and the sample. Because the force resolution of the AFM is so sensitive (0.1–1 pN), it is a powerful tool for probing the inter- and intra-molecular forces between two substances. Researchers have taken advantage of this sensitivity to quantify fundamental forces between a sample and some substance linked to the AFM cantilever or tip [41]. The AFM has enabled some truly remarkable advances in the physical sciences due to the sensitivity and ranges of force it can measure. A few examples will be discussed here. A basic understanding of the forces between the AFM tip and the sample is essential for a proper use of the instrument and the analysis of the data. A variety of forces that come into play between the tip and the sample are summarised in Table 7.2. The discussion that follows will focus on contact-mode AFM, which is the most commonly used imaging mode. A recent review highlights the effect of surface forces on dimensional measurements [30].

The total force between the tip and the sample results from the sum of various attractive and repulsive forces. As a model, consider the Lennard-Jones potential, which describes the change in inter-molecular potential energy (ϕ) that occurs as two particles, such as atoms or molecules (on tip and sample), are brought closer together. The model gives

$\varphi =4\epsilon \left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right],$ (7.1)

(7.1)

where σ is approximately the atomic or molecular diameter (distance of closest approach), ɛ is the minimum value of the potential energy, or the depth of the potential energy well, and r is the separation distance [42]. As the particles are brought closer together from relatively distance separations, the (1/r)⁶ term (i.e. Van der Waals term) describes the slow change in attractive forces. As the particles are brought even closer together, the (1/r)¹² term describes the strong repulsion that occurs when the electron clouds strongly repel one another.

The Van der Waals interaction forces are long-range, relatively weak attractive forces. The origin of the Van der Waals forces is quantum mechanical in nature; they result from a variety of interactions, primarily induced dipole and quadrupole interactions. The Van der Waals forces are non-localised, meaning that they are spread out over many atoms. Van der Waals forces for a typical AFM have been estimated to be of the order of 10–20 nN [43].

The so-called atomic force (a result of the Pauli exclusion principle) is the primary repulsive force at close approach. The magnitude of this force is difficult to predict without a detailed understanding of surface structure.

Several additional forces or interactions must be considered for an AFM tip and sample surface. Capillary adhesion is an important attractive force during imaging in air. The capillary force results from the formation of a meniscus made up of water and organic contaminants adsorbed on to the surface of the tip and the sample [36] (Figure 7.7). The capillary force has been estimated to be of the order of 100 nN or greater. When the tip and the sample are completely immersed in liquid, a meniscus does not form and the capillary forces are absent. Some tips and samples may have hydrophobic properties, in which case hydrophobic interactions must also be taken into consideration.

Water near hydrophilic surfaces is structured [34]. When the tip and the sample are brought into close contact during force microscopy in solution or humid air, repulsion arises as the structured water molecules on the surfaces of the tip and the sample are pushed away. In aqueous solutions, electrical double-layer forces, which may be either attractive or repulsive, are present near the surfaces of the tip and the sample. These double-layer forces arise because surfaces in aqueous solution are generally charged.

Lateral frictional forces must also be taken into account as the sample is scanned beneath the tip. At low forces, a linear relationship should hold between the lateral force and the force normal (vertical) to the surface with a proportionality constant equal to the coefficient of friction. This relationship is valid up to an approximately 30 nN repulsive force [44]. Frictional forces vary on an atomic scale and with temperature, scan velocity, relative humidity and tip and sample materials.

7.3.8 Tip–sample distance measurement

To obtain the distance or separation part of the force–distance curve, a point of contact (i.e. zero separation) must be defined and the recorded piezoelectric scanner position (i.e. displacement) must be corrected by the measured deflection of the cantilever. Simply adding or subtracting the deflection of the cantilever to the movement of the piezoelectric scanner determines the displacement. For example, if the sample attached to the piezoelectric scanner moves 10 nm towards the cantilever, and the cantilever is repelled 2 nm due to repulsive forces, then the actual cantilever–sample separation changes by only 8 nm. The origin of the distance axis (the point of contact) is chosen as the beginning of the region of constant compliance, that is the point on the force curve where cantilever deflection becomes a linear function of piezoelectric scanner displacement (see Figure 7.6). Just as it was difficult to convert photodiode voltage to displacement units for soft, deformable materials, it is not always easy to select the point of contact because there is no independent means of determining cantilever–sample separation. For deformable samples, the cantilever indents into the sample such that the region of constant compliance may be non-linear and the beginning point cannot be easily defined. Recently, researchers have developed an AFM with independent measurement of the piezoelectric scanner and the cantilever displacements [51].

7.3.9 Challenges and artefacts in AFM force measurements

There are a number of artefacts that have been identified in force curves. Many of these artefacts are a result of interference by the laser, viscosity effects of the solution or elastic properties of soft samples. When the sample and the cantilever are relatively remote from each other, such that there is no interaction, the force curve data should be a horizontal line (i.e. the region of non-contact; see Figure 7.6). However, the laser has a finite spot size that may be larger than the size of the cantilever such that the laser beam reflects off the sample as well as the cantilever. This is particularly troublesome for reflective substrates, often resulting in optical interference, which manifests itself as a sinusoidal oscillation or as a slight slope in the non-contact region of the force curve [52]. This affects the way in which attractive or repulsive forces are defined. A simple solution is to realign the laser on the cantilever such that the beam does not impinge upon the underlying sample. Alternatively, the oscillation artefact may be removed from the force curve with knowledge of the wavelength of the laser. This optical problem has been largely solved in commercial AFMs by using superluminescent diodes, which possess high optical power and low-coherence length.

A further artefact is the hysteretic behaviour between the approach and retraction curves in the non-contact area. The approach and retraction curves often do not overlap in high-viscosity media due to fluid dynamic effects [53]. Decreasing the rate at which the piezoelectric scanner translates the samples towards and away from the cantilever can help to minimise hysteresis by decreasing the drag caused by the fluid.

Another frequently observed artefact in the force curve is caused by the approach and retraction curves not overlapping in the region of contact, but rather being offset laterally. Such artefacts make it difficult to define the point of contact, which is necessary to obtain separation values between the sample and the tip. Such hysteresis artefacts are due to frictional effects as the tip (which is mounted in the AFM at an angle of typically 10°–15° relative to the sample) slides on the sample surface. This hysteresis is dependent upon the scan rate and reaches a minimum below which friction is dominated by stick-slip effects, and above which friction is dominated by shear forces. This artefact may be corrected by mounting the sample perpendicular to the cantilever, thereby eliminating lateral movement of the cantilever on the sample.

Viscoelastic properties of soft samples also make it difficult to determine the point of contact and to measure accurately the forces of adhesion. When the cantilever makes contact with a soft sample, the cantilever may indent the sample such that the region of contact is non-linear. It is then difficult to determine the point at which contact begins. The rate at which the sample approaches or retracts from the tip also affects the adhesive force measured on soft samples. This is because the tip and sample are weakly joined over a large contact area that does not decouple fast enough as the tip is withdrawn at very high scan rates. Thus, the membrane deforms upward as the tip withdraws, causing an increased force of adhesion. Contact between a soft sample and the tip also affects the measured adhesion force in other ways. As a tip is driven into a soft sample, the contact area increases as the sample deforms around the tip. Hence, increasing the contact force between the tip and the sample increases the contact area, which in turn increases the number of interactions between the tip and the sample. Therefore, increasing contact force results in an increased adhesive force between the tip and the sample. To compare measured adhesion values, the contact force should be selected such that it does not vary from experiment to experiment. Additionally, slow scan rates should be used to allow the tip and sample to separate during retraction.

7.4.1 Thermal measurement

Scanning thermal microscopy (SThM) uses micromachined thermal sensors integrated in an atomic force cantilever. The SThM probe can be used as either a resistive thermometer or a resistive heater. For measurement purposes, these probes are often resistive thermometers and the output is usually measured using a Wheatstone bridge.

Two main modes can be used:

SThM is capable of imaging externally generated heat sources with nanoscale resolution, but relatively poor accuracy, and macroscopic uniform temperature accurately. However, the ability of SThM to map and measure the thermal conductivity of materials has been limited to polymers or similar materials possessing low thermal conductivity in the range from 0.1 to 1 W mK⁻¹ with lateral resolution on the order of 1 µm.

7.4.2 Electrical resistivity measurement

The only method developed so far to measure accurate nanoscale resistivity is scanning spreading resistance microsocopy (SSRM). SSRM works on silicon only and relies on the existence of a series of calibration samples to relate the measured spreading resistance to the local resistivity of an unknown sample. Only a few academic papers have claimed to measure resistivity, but the uncertainty on the real area of contact between the tip and the sample and the mean-free path of the electron in the material severely limits the accuracy of the technique to, at best, an order of magnitude of the real value.

When resistance, rather than resistivity is measured, the uncertainty is below 2 % and is dominated by the local temperature drift during the duration of the scan, which creates a current offset drift in the current amplifier used in this technique.

7.6.1 Scanning electron microscopy

The scanning electron microscope (SEM) uses a very fine beam of electrons, which is made to scan the specimen under test as a raster of parallel contiguous lines (see Refs. [57,58] for thorough descriptions of electron microscopy). Upon hitting the specimen, electrons will be reflected (backscattered electrons) or generated by interaction of the primary electrons with the sample (secondary electrons). The specimen is usually a solid object, and the number of secondary electrons emitted by the surface will depend upon its topography or nature. These are collected, amplified and analysed before modulating the beam of a cathode ray tube scanned in sympathy with the scanning beam. The image resembles that seen through an optical lens but at a much higher resolution.

The dimensions of the probe beam determine the ultimate resolving power of the instrument. This is controlled in turn by the diffraction at the final aperture. The ultimate probe size in an SEM is limited by diffraction, chromatic aberration and the size of the source.

Typical SEMs can achieve image magnifications of 400 000× and have a resolution of around 1 nm with a field emission system and an in-lens detector. The magnification of the system is determined by the relative sizes of the scan on the recording camera and of the probe on the specimen surface. The magnification is, therefore, dependent upon the excitation of the scan coils, as modified by any residual magnetic or stray fields. It also depends sensitively on the working distance between the lens and the specimen. It is not easy to measure the working distance physically, but it can be reproduced with sufficient accuracy by measuring the current required to focus the probe on the specimen surface.

The camera itself may not have a completely linear scan, so distortions of the magnification can occur. In considering the fidelity of the image, it is assumed that the specimen itself does not influence the linear response of the beam; in other words that charging effects on the specimen surface are negligible. If calibration measurements of any accuracy are to be made, any metal coating employed to make the surface conducting should be very thin compared to the structure to be measured and is best avoided altogether if possible.

Since charging is much more serious for low-energy secondary electrons than for the high-energy backscattered electrons, it is preferable to use the backscattered signal for any calibration work, if the instrument is equipped to operate in this mode. For similar reasons, if the specimen is prone to charging, the use of a low-voltage primary beam, rather than an applied conductive coating, is much to be preferred, but the resolution is lost again. The indicated magnification shown on the instrument is a useful guide but should not be relied upon for accuracy better than ±10 %.

In all forms of microscopy, image degradation can occur from a number of factors. These include poor sample preparation, flare, astigmatism, aberrations, type and intensity of illumination and the numerical apertures of the condenser and objective lenses [59].

Electron backscattered diffraction (EBSD) provides crystallographic orientation information about the point where the electron beam strikes the surface [60]. EBSD has a spatial resolution down to 10–20 nm depending on the electron beam conditions that are used. Because of the unique identification of crystal orientation with grain structure, EBSD can be used to measure the size of grains in polycrystalline materials and can also be used to measure the size of crystalline nanoparticles when these are sectioned. As EBSD relies on the regularity of the crystal structure, it can also be used to estimate the degree of deformation in the surface layers of a material.

7.6.2 Transmission electron microscopy

The transmission electron microscope (TEM) operates on the same basic principle as a light microscope but uses electrons instead of light. The active components that compose the TEM are arranged in a column, within a vacuum chamber. An electron gun at the top of the microscope emits electrons that travel down through the vacuum towards the specimen stage. Electromagnetic electron lenses focus the electrons into a narrow beam and direct it onto the test specimen. The majority of the electrons in the beam travel through the specimen. However, depending on the density of the material present, some of the electrons in the beam are scattered and are removed from the beam. At the base of the microscope, the unscattered electrons hit a fluorescent viewing screen and produce a shadow image of the test specimen with its different parts displayed in varied darkness according to their density. This image can be viewed directly by the operator or photographed with a camera.

The limiting resolution of the modern TEM is of the order of 0.05 nm with aberration-corrected instruments. The resolution of a TEM is normally defined as the performance obtainable with an ‘ideal’ specimen, that is one thin enough to avoid imposing a further limit on the performance due to chromatic effects. The energy loss suffered by electrons in transit through a specimen will normally be large compared to the energy spread in the electron beam due to thermal emission velocities, and large also compared to the instability of the high-voltage supply to the gun and the current supplies to the electron lenses.

In general, the specimen itself causes loss of definition in the image due to chromatic aberration of the electrons, which have lost energy in transit through it. A ‘thick’ specimen could easily reduce the attainable resolution to 1.5–2 nm [62]. For nanoparticles, this condition could occur if a particle preparation is very dense; a good preparation of a well-dispersed particle array on a thin support film would not in general cause a serious loss in resolution.

7.6.3 Traceability and calibration of TEMs

As for SEM, the calibration factor for a TEM is the ratio of the measured dimension in the image plane and the sample dimension in the object plane. Calibration should include the whole system. This means that a calibration artefact of known size in the object plane is related to a calibration artefact of known size in the image plane. For example, the circles on an eyepiece graticule, the ruler used to measure photographs and the number of detected pixels in the image analyser should all be related to an artefact of known size in the object plane.

The final image magnification of a TEM is made up of the magnifications of all the electron lenses, and it is not feasible to measure the individual stages of magnification. Since the lenses are electromagnetic, the lens strength is dependent not only on the excitation currents, but also on the previous magnetic history of each circuit. It is essential, therefore, to cycle each lens in a reproducible manner if consistent results are to be obtained. Suitable circuitry is now included in many instruments; otherwise, each lens current should be increased to its maximum value before being returned to the operating value in order to ensure that the magnetic circuits are standardised. This should be done before each image is recorded. The indicated magnification shown on the instrument is a useful guide but should not be relied upon for an accuracy better than ±10 %.

7.6.4 Electron microscopy of nanoparticles

Electron microscopy produces two-dimensional images. The contrast mechanism is based on the scattering of electrons. Figure 7.8(a) shows a typical TEM image of gold nanoparticles. Many microscopes still record the images on photographic film. In this case, the images have to be scanned into a computer file to be analysed. However, CCD cameras are becoming increasingly popular. In this case, the image is transferred directly onto a computer file.

Traditionally, size measurements from electron microscope images are achieved by applying threshold intensity uniformly across the image. Image intensities above (or below) this level are taken to correspond to areas of the particle being measured. This is demonstrated in Figure 7.8(b), where a threshold was applied to identify the particles. Simple analysis allows the area and radius of the particle to be determined. In the case of non-spherical particles, the diameter is determined by the fitting of an ellipsoid. A histogram of the sizes can then be easily determined (Figure 7.8(c)).

Although the threshold method described above is a simple, well-defined and recognised method, it does suffer from some significant drawbacks. The first is setting the threshold level, which is difficult for poorly contrasting particles (such as small polymer particles or inhomogeneous particles) (Figure 7.9). The second, more important, drawback occurs when analysing agglomerated particles. With no significant intensity difference between the particles, a simple threshold is insufficient to distinguish between the particles and hence accurately determine size. It is usually recommended to use a watershed method.

The determination of the particle boundaries is the essential requirement for precise particle size measurement using electron microscopy. Because manual measurement is both tedious and a source of considerable non-reproducible errors, digital image processing is preferred whenever possible. Thresholding techniques are often used to separate objects from the background (see Ref. [66] for a review). Thresholding techniques rely, however, only on the distribution of grey scale values across the image, and the resulting size distribution is highly dependent on the chosen thresholding algorithm [67].

To reduce uncertainties, other information, such as the image acquisition parameters and details of the scattering process, can be used. For size measurements, threshold levels of the nanoparticles can be calculated using Monte Carlo simulations of the image formation process. The signal level of the transmitted electrons at the particle edge is then calculated by taking into account all relevant parameters of the instrument (e.g. electron energy, probe diameter, detector acceptance angle and energy sensitivity) and of the specimen (material, density, estimated particle size). Frase et al. [68] give an overview of programme packages used for SEM. Some of these packages are also able to model transmitted electrons and their detection.

Once the nanoparticles are separated from the background, a particle analysis routine [69] can be used to calculate the desired diameter; equivalent spherical, Feret or other. When image analysis is varied automatically, artefacts such as dried chemicals or touching particles are falsely included as identified objects. Depending on the size of the data set, their effect may be removed by hand or automatically via limits of some geometrical parameters, limits for the minimum and maximum size or circularity when analysing nearly spherical particles. It is recommended that such limits are set with great care and to verify that they do not alter the size distribution significantly. The use of watershed algorithms, which work by assuming that an image is a topographical surface and then modelling the flow of water upon this surface [70], often lead to systematic underestimation of the size. If high precision measurements are required, it is, therefore, advised that touching and overlapping particles should not be included in the measurement.