Scanning Electron Microscopy

Scanning Electron Microscopy (SEM) is an extremely useful technique used to obtain high-resolution images and detailed information of a samples’ surface. The typical resolution of SEM instruments ranges from < 1 nanometer to several nanometers. SEM uses a focused beam of electrons to examine the surface of a sample and analyzes the different signals emitted in a detector.

The following is an illustration of how an SEM instrument works:

Image Source: https://www.technologynetworks.com/analysis/articles/sem-vs-tem-331262

1) Electron generation. 

A high-voltage electron gun (typically 1 keV – 30 keV) generates and accelerates a beam of electrons, typically using thermionic or field emission.

• Thermionic emission involves heating a thin tungsten filament to a high temperature (approximately 2,500°C), causing electrons to be emitted. 
• Field emission is an advanced technology used to capture the microstructure image of a material. It is usually performed in a high vacuum because gas molecules tend to disturb the electron beam and the emitted secondary and backscattered electrons used for imaging. Field emission produce electron beams with the highest brightness and coherence which allows for the highest resolutions to be obtained.

2) Electron beam focusing

The electron beam generated is then focused on the sample using a series of electromagnetic lenses. These lenses are similar to optical lenses, but instead of glass lenses, they use magnetic fields to bend the electron beam.

Condenser lenses are used for this and are carefully aligned to ensure that the beam is focused to a small spot on the sample surface.

3) Scanning

The focused electron beam is scanned across the sample surface in a raster pattern, similar to how a television screen is scanned. 

This is achieved by using scanning coils that deflect the beam in the x and y directions.

4) Interaction of electron beam and the sample

As the electron beam interacts with the sample, it results in a variety of signals being emitted. These signals include:

• Secondary electrons: Low-energy electrons that are emitted from the sample as a result of the primary electron beam striking the surface.

• Backscattered electrons: High-energy electrons that are reflected back from the sample. These are high-energy electrons used to obtain high-resolution images that show the distribution of various elements that make up a sample.

• Characteristic X-rays: X-rays that are emitted when the primary electrons knock out inner-shell electrons from the atoms in the sample. The energy difference between two electron shells is equal to the characteristic X-ray energy. Characteristic X-rays that result from inelastic scattering allow for exact identification of elements present in the sample.

• Auger electrons: These are emitted at discrete energies that are characteristic of the elements present on the sample surface.  The characteristic energies of the Auger electrons are such that only the electrons from the outer 0.5 to 5 nm of the sample can escape and be detected.

5) Detection of emitted signal

The emitted signals are detected using various detectors, such as:

• Secondary electron detector: Detects secondary electrons, which are used to create an image of the sample's surface topography.

• Backscattered electron detector: Detects elastically scattered electrons, which can provide information about the sample's composition.

• X-ray detector: Detects characteristic X-rays, which can be used to determine the elemental composition of the sample.

• Auger electron spectrometer: Detects Auger electrons, which can be used to analyze the surface composition of the sample.

6) Formation of image

The detected signals are processed by a computer to create an image. The brightness of each pixel in the image corresponds to the intensity of the detected signal at that point on the sample. 

SEM images can be displayed in various modes, such as bright field, dark field, and elemental mapping.

7) Analysis of image

The SEM images can be analyzed to deduce the quantitative information about the sample, like size, shape, and distribution of features. 

This can be done using image analysis software or by manually measuring features on the image.

Application of SEM

a) Materials Science

• Microstructure analysis: SEM is widely used to study the microstructure of materials like metals, ceramics and polymers. By examining the arrangement of grains, defects, and inclusions, researchers can gain insights into the material's properties and behavior. 
• Failure analysis: When a material or component fails, SEM can be used to investigate the root cause of the failure. By examining the fracture surface, researchers can identify the type of fracture (e.g., brittle, ductile, fatigue) and identify potential defects or flaws that may have contributed to the failure.

b) Biology and Medicine

• Cell imaging: SEM can provide high-resolution images of cells and their organelles, allowing researchers to study their structure, function, and interactions. For example, SEM can be used to examine the surface morphology of bacteria, the structure of viruses, and the ultrastructure of tissues.
• Tissue analysis: SEM can be used to analyze the structure and composition of tissues, such as skin, bone, and muscle. This can be helpful for diagnosing diseases, studying the effects of treatments, and understanding biological processes.

c) Nanotechnology

• Characterization of nanomaterials: SEM is a valuable tool for characterizing nanomaterials, such as nanoparticles, nanotubes, and nanowires. It can be used to measure the size, shape, and distribution of nanomaterials, as well as to study their surface morphology and internal structure.

d) Other Fields

• Forensic science: SEM can be used for various forensic applications, such as analyzing fingerprints, examining crime scene evidence, and identifying fibers or hairs.

• Semiconductor manufacturing: SEM is essential for quality control in the semiconductor industry, as it can be used to inspect the surface features of wafers and devices.

• Environmental science: SEM can be used to study the morphology and distribution of pollutants in the environment, such as particulate matter and microplastics.

Advantages of SEM

• High Resolution: SEM can provide images with extremely high resolution, allowing for detailed examination of even the smallest structures.

• Depth of Field: SEM has a large depth of field thus it can focus on a wide range of depths simultaneously, making it ideal for imaging rough or uneven surfaces.

• Three-Dimensional Imaging: SEM can create 3D images of samples, providing valuable insights into their structure and morphology.

• Versatile Applications: SEM can be used to study a wide range of materials and samples, from metals and ceramics to biological tissues and nanomaterials.

• Quantitative Analysis: SEM can be used to obtain quantitative information about samples, such as the size, shape, and distribution of features.

Disadvantages of SEM

• Sample Preparation: SEM requires careful sample preparation, often involving coating the sample with a conductive material to prevent charging. This can be time-consuming, expensive and may introduce artifacts.
• Vacuum Environment: SEM operates in a vacuum environment, which can limit the types of samples that can be examined. Some samples may degrade or be altered under vacuum conditions.
• Cost: SEM equipment can be expensive to purchase and maintain.
• Image Interpretation: Interpreting SEM images can be challenging, especially for complex samples or when dealing with unfamiliar materials.
• Limited Penetration Depth: SEM is primarily a surface imaging technique, with limited penetration depth. This can be a limitation for studying the internal structure of thick or opaque samples.

FURTHER READING

• Goldstein, D., & Newbury, D. (2012). Scanning electron microscopy: A beginner's guide. Springer Science & Business Media.
• Scott, J. F., & Joy, D. N. (2017). Scanning electron microscopy: Techniques and applications. Springer Science & Business Media.
• Scott, J. F., & Joy, D. N. (2019). Practical scanning electron microscopy. Springer Science & Business Media.

Entropy Changes of Matter

Let us consider a closed system undergoing a reversible process with zero shaft work, the first law of thermodynamics leads us to:

Since entropy is a state function (depends only on the initial and final states), equation 3 holds true for any kind of process and is valid for all substances. 

The same can be done for enthalpy:

Equation 6 is similarly valid for all substances.

a) Entropy Change of an Ideal Gas

For an ideal gas, the following equations can be considered:

Let us now consider the enthalpy of an ideal gas.

b) Entropy Change of a Solid or a Liquid

For solids and liquids, density is almost independent of pressure and temperature. Thus, eq. 3 reduces to:

Example of Surface Tension and Capillary Action

 Example 1

a) Develop an expression to calculate the pressure inside a droplet of water,

b) Determine the pressure inside a 1cm diameter (D = 1cm) water droplet exposed to atmospheric pressure (P₀ = 101,300 N/m²) {Assume water surface tension, σ = 71.97 * 10^-3 N/m)

Solution

a) The difference in pressure on the inside and the outside tend to expand the droplet and the surface tension constrains the pressure.

Carrying out a force balance on the droplet we obtain:

(Pressure difference) x (Cross sectional area) = (Surface tension) x (Surface length)

(P_i – P₀)πR² = σ(2πR)

simplifying the above expression we get:

b) From the above equation, we can substitute the known variables to solve for the unknown Pressure inside the droplet, P_i  (where: R = D/2 = 0.5cm = 0.005m)

P_i = (101,300 N/m²) + ((2*71.97 * 10^-3 N/m)/0.005m)

P_i = 101,330 N/m²

Example 2

A capillary tube has a very small inner diameter and when it is immersed slightly in a liquid, the liquid will rise within the tube to a height that is proportional to its surface tension. This phenomenon is referred to as a capillary action.

The figure below depicts a glass capillary tube slightly immersed in water. The water rises by a height, h, within the tube and the angle between the meniscus and glass tube is θ. Perform a force balance on the system and develop a relationship between the capillary rise ,h, and surface tension ,σ.

Solution

Let us consider a cross section of the liquid column with a height h and diameter 2R with the weight of the column, W and the surface tension force, T. 

The weight can be calculated as:

The force due to surface tension acts at the circumference where the liquid touches the wall. The surface tension force is, therefore, the product of surface tension and circumference:

T = (2πR)σ

Addition of the forces in the vertical direction show that the vertical component of the surface tension force must equal the weight of the liquid column:

W = T cos(θ) or ρπR²hg = 2πRσcos(θ)

Thus, rearranging we can solve for the height, h.

 

Properties of Fluids

Fluid mechanics equations helps to predict the behavior of fluids in various flow situations. A fluid can be defined as a substance that deforms continuously under an applied shear stress.

Several properties need to be provided in order to use these equations. Some of these properties include: viscosity, pressure, density, kinematic viscosity, surface tension, specific heat, internal energy, enthalpy, and compressibility.

a) Viscosity

This is one of the most important fluid properties. Viscosity can be defined as the measure of resistance a fluid has to an externally applied shear stress.

Let us consider a fluid in between two horizontal parallel plates as shown below:

The upper plate has a contact area, A, with the fluid and is being pulled to the right by a force, F₁ at a velocity, V₁. The measurement of velocity at infinitesimally small distances along the fluid results in a velocity distribution graph as illustrated in the figure above. Since the fluid near the moving surface adheres to it, the velocity of the surface (x=Δy) is V₁. This is referred to as the nonslip condition. When x=0 (at the stationary plate) velocity is zero (again due to nonslip condition). The slope of the velocity distribution is dV₁/dy.

If a different force F₂ is applied on this system, a different slope (strain rate) of dV₂/dy is obtained. Thus, we can summarize that each applied force yields to only one shear stress and one strain rate.

Plotting τ vs dV/dy for a fluid like water results in the following graph:

As can be seen, the points lie on a straight line that passes through the origin and the slope of this line is the viscosity of the fluid as it is a measure of the fluid’s resistance to shear. i.e., viscosity indicates how a fluid will react (dV/dy) under the action of an external shear stress (τ). The above graph is characteristic of a Newtonian fluid. A Newtonian fluid is simply a fluid whose viscosity is not affected by shear rate at constant temperature. Water, oil, and air are examples of Newtonian fluids.

Since there are also non-Newtonian fluids as well, if we plot τ vs dV/dy we obtain the following rheological diagram.

Newtonian fluids obey Newton’s law of viscosity that is represented as the equation below, any fluid that doesn’t comply with this equation can be classified as a non-Newtonian fluid.

Where:

• τ = the applied shear stress (lbf=ft² or N=m²)
• μ = the absolute or dynamic viscosity of the fluid (lbf.s/ft² or N.s/m²)
• dV/dy = the strain rate (rad/s)

The dynamic viscosity is simply the resistance to movement of one fluid layer over another.

Non-Newtonian fluids are divided into three categories: time-independent, time-dependent, and viscoelastic.

    (i) Time-Independent Fluids

These fluids are divided into dilatant, pseudoplastic and Bingham plastics.

Dilatant fluids exhibit an increase in viscosity with an increase in shear stress. Examples include wet beach sand and other water solutions containing a high concentration of powder.  

A power law equation (called the Ostwald–deWaele equation) usually gives an adequate description:

where: K is called a consistency index (lbf.sⁿ/ft² or N.sⁿ/m²) and n is a flow behavior index.

Pseudoplastic fluids exhibit a decrease in viscosity with an increase in shear stress. Examples include mayonnaise, greases and starch suspensions. A power law equation also applies:

Bingham plastic fluids behave as solids until an initial yield stress τ₀ is exceeded. Beyond τ₀, Bingham plastics behave like Newtonian fluids. Examples include toothpaste, soap, paint, chocolate mixtures and paper pulp. The descriptive equation is:

   (ii) Time-Dependent Fluids

These are divided into rheopectic and thixotropic fluids.

In a rheopectic fluid, a shear stress that increases with time gives the rheopectic fluid a constant strain rate.

Thixotropic fluid behaves opposite to rheopectic fluids i.e., a shear stress that decreases with time gives a thixotropic fluid a constant strain rate.

   (iii) Viscoelastic Fluids

These are fluids that show both elastic and viscous properties, they partly recover elastically from deformations cased during flow. Examples include flour dough.

b) Pressure

Pressure is defined as a normal force per unit area existing in the fluid with units lbf/ft² or N/m². Fluids, whether at rest or moving, exhibit some type of pressure variation either with height or with horizontal distance.

In a closed-conduit flow like flow in pipes, differences in pressure throughout the conduit maintain the flow. Pressure forces are significant in this regard.

c) Density

The density of a fluid is its mass per unit volume, represented as ρ.

One quantity of importance related to density is specific weight. Whereas density is mass per unit volume, specific weight is weight per unit volume (lbf/ft³ or N/m³). Specific weight is related to density by:

SW = ρg

Another useful quantity related to density of a substance is specific gravity. The specific gravity of a substance is the ratio of its density to the density of water at 4°C:

d) Kinematic Viscosity

Kinematic viscosity, ν, with the units ft²/s or m²/s is the ratio of absolute viscosity to density:

e) Specific Heat

The specific heat of a substance is the heat required to raise a unit mass of the substance by 1°. The dimension of specific heat is energy/(mass.temperature).

The process by which the heat is added also makes a difference, especially for gases. The specific heat for a gas undergoing a process occurring at constant pressure involves a different specific heat than that for a constant volume process.

f) Internal Energy

This is the energy associated with the motion of the molecules of a substance. For example, a gas can have three types of energy:

• energy of position (potential energy),
• energy of translation (kinetic energy) and
• energy of molecular motion (internal energy).

The added heat does not increase the potential or kinetic energies but instead affects the motion of the molecules. This results in an increase in temperature. For a perfect gas with constant specific heats, internal energy is:

ΔU = c_vΔT

where ΔU is a change in internal energy per unit mass with dimensions of energy/mass. 

g) Enthalpy 

This is the total heat energy in a system equivalent to the sum of total internal energy and resulting energy due to its pressure and volume. For a perfect gas with constant specific heats, it can be shown that:

ΔH = c_pΔT

Similar to internal energy, enthalpy has the dimension of energy/mass.

h) Surface Tension

This is a measure of the energy required to reach below the surface of a liquid bulk and bring molecules to the surface to form a new area. Therefore, surface tension arises from molecular considerations and applies only for liquid–gas or liquid–vapor interfaces.

For a liquid-gas interface, attraction of molecules is due to cohesive forces. If the liquid is in contact with a solid, adhesive forces can also be considered. Remember, cohesive forces are the forces that attract molecules of the same substance whereas, adhesive forces binds molecules of different substances together.

The dimension of surface tension is energy/area (lbf/ft or N/m) and is generally denoted by the symbol, σ.

i) Bulk Modulus or Compressibility Factor

All materials, whether solids, liquids or gases, are compressible. This means that for a given volume, V of a given mass, will be reduced to V – dV when a force is exerted uniformly over its entire surface. If the force per unit area of surface increases from P to P + dP, the relationship between change of pressure and change of volume depends on the bulk modulus of the material.

Compressibility factor is expressed as follows:

The isothermal bulk modulus is the reciprocal of the compressibility factor and expressed as follows:

Since the dimensions of volume change and volume are the same, the denominator of the above equation is dimensionless. Thus the isothermal bulk modulus has the same units as pressure.

The isothermal bulk modulus can be measured experimentally, but it is more convenient to calculate it using data on the velocity of sound in the medium. The sonic velocity in a liquid or a solid is related to the isothermal bulk modulus and the density by the following equation:

Entropy Balance

Entropy is an extensive property, like mass and energy, that can be used to characterize the state of thermodynamic systems at equilibrium. The entropy of a system changes as a result of a process but it is not a conserved quantity. Generation of entropy, S_gen, is equal to the entropy change of the universe, i.e.,

S_gen = ΔS_universe

         =ΔS_system + ΔS_surroundings  ≥  0               (1)

ΔS_system may be less than zero. However, ΔS_universe is either zero (reversible process) or greater than zero (irreversible process).

For an open system (both mass and energy is exchanged with surroundings), the rate of entropy generation is expressed as:

Ṡ_gen = Ṡ_system + Ṡ_surrounding              (2)

The rate of entropy change of the system is expressed as:

The entropy of the surroundings may change through two ways: (i) mass flow, (ii) heat transfer. Note that entropy transfer has nothing to do with work. Thus,

This is also known as the entropy balance. Note that dQ_surr = − dQ_sys and dS_gen ≥ 0.

Simplifying Eq. 6 for different types of systems is given below:

Isolated Systems

Since there is no exchange of mass and energy between the system and its surroundings, Eq. 6 simplifies to:

Closed System

Since there is no exchange of mass between the system and its surroundings, i.e., d_min = dm_out = 0, Eq. 6 simplifies to:

Steady State System

Noting that:

d_min = dm_out = dm                         and

                                                                   d(mŜ)_sys = 0

 Equation 6 simplifies to:

Entropy Change of the Universe

The entropy change of the universe is the sum of the entropy change of the system and the entropy change of the surroundings:

ΔS_universe = ΔS_system + ΔS_surrounding

To determine the entropy change of the universe, let us consider the following processes:

• Adiabatic expansion

• Adiabatic compression

• Heat Engine

• Heat Pump

Adiabatic expansion

Let us consider a reversible adiabatic expansion of a gas in a piston-cylinder device from P₁ to P₂. Since work is done by the system, the gas temperature decreases to the final state temperature T₂. The process is represented on a P – T diagram below:

During expansion, work is done by the system and -W is a positive quantity. As the initial and final state pressures are the same, part of the work done by the system during the irreversible expansion process is used to overcome friction and as a result:

This indicates that the irreversible process leads to a smaller decrease in internal energy than the reversible process. Equation 3 can be expressed as:

U₁ – U₂ > U₁ – U₃    thus U₃ > U₂                        (4)

Therefore T₃ > T₂. It is possible to go from state 3 to state 2 by removing heat reversibly to decrease the temperature. The change in entropy for this process is given by:

Since heat is removed from the system, i.e., dQ < 0, then S₃ > S₂.

On the other hand, the reversible process occurs at constant entropy, (S₁ = S₂), and thus S₃ > S₁. Note that the processes 1 → 2 and 1 → 3 cause no entropy change of the surroundings as they both are adiabatic. Hence, the system entropy change is the entropy change of the universe. The universe entropy changes due to the reversible and irreversible adiabatic expansion processes are summarized as follows:

Adiabatic Compression

Let us consider a reversible adiabatic compression of a gas in a piston-cylinder device from P₁ to P₂. Since work is done on the system, the gas temperature will increase to the final state temperature, T₂. The process is represented on a P – T diagram as follows:

During compression, work is done on the system and W is positive. Since the initial and final state pressures are the same, the irreversible compression requires an extra work to overcome friction and as a result:

This indicates that the irreversible process leads to a larger increase in internal energy than the reversible process. Equation 8 can then be expressed as:

U₃ – U₁ > U₂ – U₁    thus U₃ > U₂                        (9)

Therefore T₃ > T₂. It is possible to go from state 3 to state 2 by removing heat reversibly to decrease the temperature. The change in entropy for this process is given by:

Since heat is removed from the system, i.e., dQ < 0, then S₃ > S₂.

On the other hand, the reversible process occurs at constant entropy, (S₁ = S₂), and thus S₃ > S₁. These processes have no entropy change of the surroundings as they both are adiabatic. Hence, the system entropy change is the entropy change of the universe. The universe entropy changes due to the reversible and irreversible adiabatic expansion processes are summarized as follows:

Heat Engine

Let us consider two heat engines that operate between heat reservoirs at the same constant temperatures T_H and T_C as shown below:

The reversible on extracts Q_H amount of heat from the high temperature reservoir and discards Q_crev amount of heat to the low temperature reservoir, with a work output of -W_rev.

The irreversible heat engine receives the same amount of heat from the high temperature reservoir and discards Q_cirrev amount of heat to the low temperature reservoir, with a work output of − W_irrev.

The entropy change of the universe is:

ΔS_universe = ΔS_H + ΔS_{heat engine} + ΔS_C                                (11)

As the heat engine operates is cyclical, there is no entropy change of the heat engine and thus Eq. 11 simplifies as:

ΔS_universe = ΔS_H + ΔS_C                                (12)

For a reversible heat engine, equation 12 can be expressed as:

For a heat engine ΔU = 0 and − W = Q. Thus,

Q_H - Q_crev = -W_rev      and   Q_H - Q_cirrev = -W_irrev              (17)

Since both of the heat engines operate between the same temperature limits, the work output of the reversible heat engine is always greater than the irreversible one, i.e.,

-W_rev > -W_irrev              (18)

Using equation 18 in equation 17 we see that:

Q_cirrev > Q_crev                   (19)

Using the identity in equation 19, we can compares equations 15 and 16 to result in:

ΔS_{universe|irrev} > 0               (20)

Heat Pump

Let us consider two heat pumps (refrigerators) that operate between the same constant temperature heat reservoirs T_H and T_C as shown: 

The reversible heat pump extracts Q_C amount of heat from the low temperature reservoir and discards Q_Hrev amount of heat to the high temperature reservoir. Thus, W_rev amount of work is done on the heat pump.

The irreversible heat pump extracts the same amount of heat from the low temperature reservoir but discards Q_Hirrev amount of heat to the high temperature reservoir while receiving W_irrev amount of work from the surroundings.

The entropy change of the universe is expressed as:

                                   ΔS_universe = ΔS_H + ΔS_{heat pump} + ΔS_C                                (21)

As mentioned previously, since the heat engine operates is cyclical, there is no entropy change of the heat engine and thus Eq. 11 simplifies as:

ΔS_universe = ΔS_H + ΔS_C                                (22)

For a reversible heat pump, equation 22 can be expressed as:

For a heat pump ΔU = 0 and − W = Q. Thus,

Q_C – Q_Hrev = -W_rev      and   Q_C – Q_Hirrev = -W_irrev       (26) 

Since both of the heat engines operate between the same temperature limits, the work received by the reversible heat engine is always less than the irreversible one, i.e.,

                                                                  W_rev <  W_irrev              (27)

Using equation 27 in equation 26 we see that:

Q_Hirrev > Q_Hrev            (28)

Since Q_Hirrev is larger than in the reversible case, Q_Hrev, comparing Equations 24 and 25 results in:

ΔS_{universe|irrev} > 0            (29)

Note:

The results for the thermal processes (Heat engine & heat pump) are the same as those obtained for the mechanical processes (Adiabatic expansion & compression) and can be summarized by the equation