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Chapter 1: Introduction to Nanomaterials

The Nanoscale World and Size Effects

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Chapter 1: Introduction to Nanomaterials

Develop an intuitive understanding of the unique physical properties and size effects that emerge at the nanoscale. Quickly grasp the classification of representative nanomaterials and their historical background.

💡 Note: "The smaller it gets, the more surface behavior dominates." Quantum confinement is easiest to grasp if you picture it as "the musical scale becoming coarser."

The Nanoscale World and Size Effects

Learning Objectives

By completing this chapter, you will be able to:

1.1 What Are Nanomaterials?

Defining the Nanoscale

The first step in understanding nanomaterials is to get a real sense of the scale that "nano" represents.

The nanometer (nm) is an extremely small unit of length equal to one billionth of a meter:

$$ 1 \text{ nm} = 10^{-9} \text{ m} = 0.000000001 \text{ m} $$

To grasp this extraordinarily small scale, let's compare it to familiar sizes:

Object Size Nanometer Equivalent
Human height About 1.7 m 1,700,000,000 nm
Hair thickness About 80 μm 80,000 nm
Red blood cell About 7 μm 7,000 nm
Bacterium (E. coli) About 2 μm 2,000 nm
Virus (influenza) About 100 nm 100 nm
Typical size of nanomaterials 1-100 nm 1-100 nm
DNA double helix diameter About 2 nm 2 nm
Water molecule About 0.3 nm 0.3 nm
Atom (carbon) About 0.15 nm 0.15 nm

Nanomaterials are materials on a scale comparable to or smaller than viruses. At this scale, anywhere from a few to several thousand atoms come together to form a single structure.

Definition of Nanomaterials

The International Organization for Standardization (ISO) defines nanomaterials in its technical specification ISO/TS 80004-1 as follows:

Nanomaterial: A material with at least one external dimension, or an internal structure, on the nanoscale (approximately 1 nm to 100 nm)

The key point in this definition is the phrase "at least one dimension." In other words, it is not necessary for all three dimensions to be nano-sized—a material is called a nanomaterial even if only one direction is nano-sized. This concept leads to the dimensional classification (0D, 1D, 2D, 3D) described later.

Nanomaterials have the following four key characteristics:

  1. Dramatic increase in surface-area-to-volume ratio: As size decreases, the proportion of atoms located on the surface increases
  2. Emergence of quantum effects: When particle size becomes comparable to the electron wavelength, quantum mechanical effects become significant
  3. Size-dependent properties: Even with the same chemical composition, color, melting point, catalytic activity, and other properties change with size
  4. Unique optical properties: Optical properties absent in bulk materials appear, such as the localized surface plasmon resonance of metal nanoparticles

Why Nanomaterials Attract Attention

Bulk materials (materials of ordinary size) and nanomaterials can exhibit completely different properties even with the same chemical composition.

As a representative example, let's look at the size effect of gold (Au):

Particle Size Color Melting Point Characteristics
Bulk Golden (yellow-gold) 1,064°C Chemically stable, no catalytic activity
50-100 nm Blue-violet About 900-1,000°C Localized surface plasmon resonance
20-30 nm Red About 700-800°C Strong light absorption, bioimaging
5-10 nm Red to purple About 500-600°C High catalytic activity
2-3 nm Variable About 300-400°C Emergence of quantum effects

Even for the same element, gold, properties change this dramatically depending on particle size. This size dependence is precisely what makes nanomaterials research so fascinating, and it is the source of a wide range of potential applications.

1.2 Size Effects and Surface/Interface Effects

Increase in Surface-Area-to-Volume Ratio

One of the most important characteristics of nanomaterials is the dramatic increase in surface-area-to-volume ratio.

As a simple example, let's consider a spherical particle of radius $r$.

$$ \frac{S}{V} = \frac{4\pi r^2}{\frac{4}{3}\pi r^3} = \frac{3}{r} $$

This equation shows that as the particle radius decreases, the surface-area-to-volume ratio increases. In other words, if the size becomes 1/10, the surface-area-to-volume ratio becomes 10 times larger.

Let's compare this with concrete numbers:

Particle Diameter Surface-Area-to-Volume Ratio Total Number of Atoms (Au) Fraction of Surface Atoms
1 cm (10⁷ nm) 0.6 m⁻¹ ~10²² <0.001%
1 mm (10⁶ nm) 6 m⁻¹ ~10¹⁹ ~0.01%
100 μm (10⁵ nm) 60 m⁻¹ ~10¹⁶ ~0.1%
10 μm (10⁴ nm) 600 m⁻¹ ~10¹³ ~1%
1 μm (1000 nm) 6,000 m⁻¹ ~10¹⁰ ~10%
100 nm 60,000 m⁻¹ ~10⁷ ~20%
10 nm 600,000 m⁻¹ ~10⁴ ~40%
5 nm 1,200,000 m⁻¹ ~10³ ~60%
2 nm 3,000,000 m⁻¹ ~250 ~80%

In a 10 nm gold nanoparticle, about 40% of all atoms are located on the surface. At 2 nm, as much as 80% of the atoms are on the surface.

This increase in surface atoms brings about the following dramatic changes in physical properties:

Effects of Surface Energy

In nanoparticles, surface energy has a major influence on the overall properties of the material.

A representative phenomenon is melting point depression. Nanoparticles melt at lower temperatures than bulk materials.

This phenomenon is known as the Gibbs-Thomson effect and can be approximated by the following equation:

$$ T_m(r) = T_{m,\text{bulk}} \left(1 - \frac{2\gamma V_m}{r \Delta H_f}\right) $$

Here: - $T_m(r)$: melting point of a particle of radius $r$ - $T_{m,\text{bulk}}$: melting point of the bulk material - $\gamma$: surface energy (surface tension) - $V_m$: molar volume - $\Delta H_f$: enthalpy of fusion - $r$: particle radius

Experimental data for melting points of gold nanoparticles:

Particle Diameter Melting Point Reduction from Bulk
Bulk 1,064°C 0°C
100 nm ~1,050°C ~14°C
50 nm ~1,020°C ~44°C
20 nm ~950°C ~114°C
10 nm ~850°C ~214°C
5 nm ~650°C ~414°C
2 nm ~350°C ~714°C

A 2 nm gold nanoparticle melts at a temperature more than 700°C lower than bulk gold. This property is utilized in the development of low-temperature sintering materials and thermally responsive materials.

Enhancement of Catalytic Activity

The increase in surface-area-to-volume ratio leads to a dramatic enhancement of catalytic activity.

Let's consider platinum (Pt) catalysts as an example:

Relationship between platinum particle size and catalytic activity:

Pt Particle Size Surface Area (per gram) Relative Catalytic Activity Cost Efficiency
Bulk plate ~1 m²/g 1× 1×
10 μm powder ~0.1 m²/g 2× 2×
100 nm powder ~10 m²/g 50× 50×
10 nm nanoparticles ~100 m²/g 500× 500×
3 nm nanoparticles ~300 m²/g 1,500× 1,500×

3 nm platinum nanoparticles exhibit 1,500 times the catalytic activity of a bulk platinum plate. This means that 1,500 times the performance can be extracted from the same mass of platinum, contributing significantly to reducing the consumption of scarce metals.

1.3 Quantum Effects and Quantum Confinement

Emergence of Quantum Effects

When particle size reaches the nanoscale, quantum mechanical effects that cannot be explained by classical physics become significant.

The key to understanding quantum effects is the de Broglie wavelength. All particles possess wave-like properties, and their wavelength $\lambda$ is given by the following equation:

$$ \lambda = \frac{h}{p} = \frac{h}{mv} $$

Here: - $h$: Planck's constant ($6.626 \times 10^{-34}$ J·s) - $p = mv$: momentum (mass × velocity) - $m$: mass of the particle - $v$: velocity of the particle

Let's calculate the de Broglie wavelength of an electron at room temperature (300 K):

$$ \lambda = \frac{h}{m_e v} \approx \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 1.17 \times 10^5} \approx 6.2 \text{ nm} $$

The de Broglie wavelength of an electron is roughly 6 nm. When particle size becomes comparable to or smaller than this wavelength, electrons behave as "confined waves" within the particle, and quantum effects become important.

Quantum Confinement Effect

The quantum confinement effect is a phenomenon in which electrons or holes become confined in a narrow space, causing their energy states to become discrete.

As the simplest model, let's consider the one-dimensional infinite square well potential. The energy levels of a particle confined in a box of length $L$ are:

$$ E_n = \frac{n^2 h^2}{8mL^2} \quad (n = 1, 2, 3, \ldots) $$

Here: - $n$: quantum number - $h$: Planck's constant - $m$: mass of the particle - $L$: length of the box (particle size)

This equation leads to several important conclusions:

  1. Energy is discrete: rather than continuous values, only specific values ($E_1, E_2, E_3, \ldots$) are allowed
  2. A minimum energy (ground state) exists: $E_1 = \frac{h^2}{8mL^2}$, which is not zero
  3. The energy gap depends on size:

$$ \Delta E = E_2 - E_1 = \frac{3h^2}{8mL^2} \propto \frac{1}{L^2} $$

The smaller the particle size, the larger the energy gap becomes.

This is why the color of semiconductor nanoparticles (quantum dots) changes with size.

Controlling Emission Color in Semiconductor Quantum Dots

Quantum dots (QDs) are semiconductor nanoparticles whose band gap (forbidden band width) changes with size, allowing control of the emission color.

Example of CdSe (cadmium selenide) quantum dots:

Particle Diameter Band Gap Emission Color Emission Wavelength Application Example
Bulk 1.74 eV Infrared ~710 nm -
10 nm 1.85 eV Red ~670 nm Red QLED
6 nm 2.00 eV Orange ~620 nm Displays
4 nm 2.25 eV Yellow-green ~550 nm Bioimaging
3 nm 2.50 eV Green ~495 nm Green QLED
2 nm 2.75 eV Blue ~450 nm Blue QLED

As particle diameter decreases from 10 nm to 2 nm, the band gap increases from 1.85 eV to 2.75 eV, and the emission color changes from red to blue.

This can be explained by the Brus equation (the simplest approximate form):

$$ E_g(r) = E_{g,\text{bulk}} + \frac{h^2}{8r^2}\left(\frac{1}{m_e^*} + \frac{1}{m_h^*}\right) - \frac{1.8e^2}{4\pi\epsilon\epsilon_0 r} $$

Here: - $E_g(r)$: band gap of a quantum dot of radius $r$ - $E_{g,\text{bulk}}$: band gap of the bulk semiconductor - $m_e^$, $m_h^$: effective masses of the electron and hole - $e$: charge of the electron - $\epsilon$: dielectric constant - Second term: energy increase due to quantum confinement ($\propto 1/r^2$) - Third term: energy decrease due to Coulomb interaction ($\propto 1/r$)

Major applications of quantum dots:

  1. QLED (quantum dot LED displays): Commercialized by Samsung, Sony, and others; color reproduction improved by 150% compared to conventional displays
  2. Bioimaging: Brighter than fluorescent dyes and more resistant to photobleaching
  3. Solar cells: Improved theoretical efficiency in multi-junction solar cells (potential to exceed the Shockley-Queisser limit)
  4. Quantum information technology: Candidate material for qubits

Localized Surface Plasmon Resonance in Metal Nanoparticles

In metal nanoparticles, a distinctive optical phenomenon known as localized surface plasmon resonance (LSPR) appears.

A plasmon is a collective oscillation of free electrons in a metal. In nanoparticles, the electric field of light causes the electron cloud to oscillate, and resonance occurs at a specific wavelength.

LSPR of gold nanoparticles:

Particle Size/Shape LSPR Wavelength Observed Color Application
10-20 nm spherical ~520 nm Red Biosensing
50 nm spherical ~530 nm Red-violet Photothermal therapy
100 nm spherical ~570 nm Blue-violet SERS substrates
Nanorods (aspect ratio 3:1) ~650 nm, ~520 nm Blue-green Imaging
Nanoshells (Au/SiO₂) ~800 nm Transparent (near-infrared) Cancer hyperthermia therapy

Application examples of LSPR:

  1. Biosensing: Antibodies are attached to gold nanoparticles, and binding of a target molecule shifts the LSPR wavelength (detection limit: on the order of pM)
  2. Surface-enhanced Raman scattering (SERS): Raman signals are enhanced by a factor of 10⁶ to 10¹⁴, enabling even single-molecule detection
  3. Cancer hyperthermia therapy: Near-infrared light (which penetrates tissue well) heats gold nanoparticles, selectively killing cancer cells
  4. Color filters: Plasmonic color filters with controlled LSPR wavelengths

1.4 Classification of Nanomaterials

Nanomaterials are classified according to how many dimensions are nano-sized.

Classification by Dimensionality

flowchart TD A[Nanomaterials] --> B[0-Dimensional 0D] A --> C[1-Dimensional 1D] A --> D[2-Dimensional 2D] A --> E[3-Dimensional 3D] B --> B1[Nanoparticles] B --> B2[Quantum Dots] B --> B3[Fullerenes] C --> C1[Carbon Nanotubes CNT] C --> C2[Nanowires] C --> C3[Nanofibers] D --> D1[Graphene] D --> D2[Transition Metal Dichalcogenides TMDCs] D --> D3[Nanosheets] E --> E1[Nanoporous Materials] E --> E2[Nanocomposites] E --> E3[Nanocrystalline Materials] style A fill:#e1f5ff style B fill:#fff4e1 style C fill:#ffe1f5 style D fill:#e1ffe1 style E fill:#f5e1ff

Classification criteria:

0-Dimensional Nanomaterials (0D)

Nanoparticles

Quantum Dots

Fullerenes

1-Dimensional Nanomaterials (1D)

Carbon Nanotubes (CNT)

Carbon nanotubes are structures formed by rolling a graphene sheet (a hexagonal lattice of carbon atoms) into a cylindrical shape.

Classification: 1. Single-walled carbon nanotubes (SWCNT): composed of a single graphene sheet - Diameter: 0.4-3 nm - Electrical properties: metallic or semiconducting depending on the chirality - Strength: tensile strength ~100 GPa (100 times that of steel)

  1. Multi-walled carbon nanotubes (MWCNT): multiple graphene sheets stacked concentrically - Diameter: 10-100 nm - Electrical properties: mainly metallic - Conductivity: higher than copper (up to 10⁷ S/m)

Key properties:

Property Value Comparison
Tensile strength 50-100 GPa 50-100 times that of steel
Young's modulus ~1 TPa Comparable to diamond
Electrical conductivity Up to 10⁷ S/m Close to copper (6×10⁷ S/m)
Thermal conductivity ~3,000 W/m·K Exceeds diamond (2,200 W/m·K)
Current density Up to 10⁹ A/cm² 1,000 times that of copper

Applications: - Composite materials: lightweight, high-strength materials (aerospace, sporting goods) - Electronic devices: CNT transistors (CNT-FET), transparent conductive films - Energy storage: lithium-ion battery electrodes, supercapacitors - Sensors: gas sensors, biosensors

Nanowires

Nanofibers

2-Dimensional Nanomaterials (2D)

Graphene

Graphene is a two-dimensional sheet approximately 0.34 nm thick (equivalent to a single layer of carbon atoms), in which carbon atoms are arranged in a hexagonal lattice. It was first isolated in 2004 by Andre Geim and Konstantin Novoselov using mechanical exfoliation, for which they were awarded the 2010 Nobel Prize in Physics.

Remarkable properties:

Property Value Comparison
Electrical conductivity ~10⁸ S/m About 100 times that of copper
Electron mobility 200,000 cm²/V·s (room temperature) More than 100 times that of silicon
Tensile strength 130 GPa About 200 times that of steel
Young's modulus 1 TPa Comparable to diamond
Thermal conductivity 5,000 W/m·K About 12 times that of copper (400 W/m·K)
Light transmittance 97.7% (single layer) Nearly transparent
Specific surface area 2,630 m²/g (theoretical) More than twice that of activated carbon

Application fields: 1. Electronics: Graphene FETs (high-speed transistors), transparent conductive films, flexible electronics 2. Energy: Lithium-ion battery electrodes (3x capacity improvement), supercapacitors (10x energy density) 3. Composite materials: Graphene/polymer composites (improved strength and conductivity) 4. Sensors: Chemical sensors, biosensors (single-molecule detection possible) 5. Transparent conductive films: Touch panels, solar cells (expected as an ITO replacement)

Transition Metal Dichalcogenides (TMDCs)

Nanosheets

3-Dimensional Nanomaterials (3D)

Nanoporous Materials

Nanocomposites

Nanocrystalline Materials

1.5 Application Fields of Nanomaterials

Nanomaterials leverage their distinctive physical properties to drive innovation across a wide range of fields.

Energy Sector

Lithium-ion Batteries (LIB)

Fuel Cells

Solar Cells

Electronics Sector

Semiconductor Devices

Displays

Medical and Biotechnology Sector

Drug Delivery Systems (DDS)

Imaging

Biosensors

Environmental and Catalysis Sector

Water Treatment

Air Purification

CO₂ Reduction

Materials and Structural Sector

Composite Materials

Coatings

1.6 Market and Future Outlook for Nanomaterials

Global Market Size

The market for nanomaterials and nanotechnology is expanding rapidly.

Market size and growth rate by sector (2023-2030 forecast):

Sector 2023 Market Size 2030 Forecast CAGR
Nanoelectronics $45 billion $85 billion 9.5%
Nanomedicine and drug delivery $38 billion $72 billion 10.2%
Nanoenergy (batteries and solar cells) $32 billion $68 billion 11.3%
Nanocomposites $27 billion $51 billion 9.8%
Nanocatalysts and environmental materials $18 billion $34 billion 9.4%
Other (coatings, cosmetics, etc.) $40 billion $70 billion 8.2%
Total $200 billion $380 billion 9.8%

Major drivers of market expansion: 1. Growing adoption of electric vehicles (EVs): demand for high-performance batteries 2. 5G/6G communications: high-frequency devices, transparent conductive films 3. Renewable energy: solar cells, fuel cells, energy storage 4. Personalized medicine: drug delivery, biosensors 5. Carbon-neutral policies: CO₂ reduction catalysts, lightweight materials

Leading Research Countries and Regions

Publication ranking (nanomaterials field, 2022):

  1. China: 72,000 papers (35%)
  2. United States: 38,000 papers (18%)
  3. India: 22,000 papers (11%)
  4. South Korea: 14,000 papers (7%)
  5. Japan: 12,000 papers (6%)
  6. Germany: 11,000 papers (5%)
  7. Iran: 9,000 papers (4%)
  8. United Kingdom: 8,000 papers (4%)

Top companies by patent applications (cumulative 2018-2022):

  1. Samsung Electronics (South Korea): 3,200 applications
  2. LG Chem (South Korea): 2,800 applications
  3. BASF (Germany): 2,100 applications
  4. IBM (United States): 1,900 applications
  5. Intel (United States): 1,700 applications
  6. Toray (Japan): 1,500 applications
  7. Panasonic (Japan): 1,400 applications
  8. 3M (United States): 1,300 applications

Future Research Trends

1. Sustainable Nanomaterials

2. Multifunctional Nanomaterials

3. Computational and AI-Driven Nanomaterial Design

4. Nano-Bio Convergence

1.7 Safety and Ethical Considerations of Nanomaterials

As nanomaterials rapidly advance, consideration of their safety and ethical dimensions has become increasingly important.

Safety Concerns

Potential risks:

  1. Biological effects - Cytotoxicity: oxidative stress from the generation of reactive oxygen species (ROS) - Pulmonary effects: inhaled nanoparticles reach deep into the lungs, risk of inflammation and fibrosis - Barrier penetration: potential to cross the blood-brain barrier and placenta - Bioaccumulation: long-term accumulation in the liver and spleen

High-risk examples: - CNT: shape similar to asbestos, risk of pulmonary fibrosis - Silver nanoparticles: accumulation in liver cells, cytotoxicity from silver ion release - Titanium dioxide (TiO₂) nanoparticles: classified by IARC (International Agency for Research on Cancer) as Group 2B (possibly carcinogenic to humans)

  1. Environmental effects - Impact on aquatic ecosystems: toxicity to algae and fish - Impact on soil microorganisms: reduced activity of nitrogen-fixing bacteria - Bioaccumulation: potential for concentration through the food chain

  2. Occupational exposure - Risks in manufacturing settings: inhalation, skin contact - Waste disposal: dispersal during incineration, leaching from landfills

Current state of risk assessment:

Regulations and Guidelines

Major regulatory bodies and regulations:

  1. European Union (EU) - REACH regulation (Registration, Evaluation, Authorisation and Restriction of Chemicals): nanomaterials are also covered, with special registration requirements - Cosmetics Regulation: mandatory labeling for products containing nanomaterials (ingredient names marked "nano") - Food regulations: approval system for novel foods containing nanomaterials

  2. United States - FDA (Food and Drug Administration): guidance on products containing nanomaterials (2014) - EPA (Environmental Protection Agency): environmental risk assessment of nanomaterials - NIOSH (National Institute for Occupational Safety and Health): recommended occupational exposure limits

  3. Japan - Ministry of Economy, Trade and Industry: guidelines on the proper management of nanomaterials - Ministry of Health, Labour and Welfare: evaluation under the Act on the Evaluation of Chemical Substances and Regulation of Their Manufacture (Chemical Substances Control Law) - Ministry of the Environment: environmental impact assessment of nanomaterials

Safe handling:

Ethical and Social Issues

1. Technology Divide

2. Transparency and Disclosure

3. Responsible Research and Innovation (RRI)

Japan's nanomaterial safety assessment projects:

Summary

In this chapter, we learned the fundamentals of nanomaterials. Here is a summary of the key points:

  1. Nanoscale: An extremely small world of 1-100 nm, about 10 times the size of a water molecule and comparable to that of a virus.

  2. Size effects: The surface-area-to-volume ratio increases dramatically (40% of atoms are on the surface at 10 nm). Changes in properties occur, including melting point depression, enhanced catalytic activity, and increased reactivity.

  3. Quantum effects: The quantum confinement effect emerges when particle size becomes comparable to the electron's de Broglie wavelength (~6 nm). In semiconductor quantum dots, the band gap and emission color can be controlled by size.

  4. Classification by dimensionality: 0-dimensional (nanoparticles, quantum dots), 1-dimensional (CNT, nanowires), 2-dimensional (graphene, TMDCs), 3-dimensional (nanoporous materials, nanocomposites). Properties and applications differ by dimension.

  5. Wide-ranging applications: These span energy (batteries, fuel cells, solar cells), electronics (transistors, displays), medicine (drug delivery, imaging), the environment (catalysts, water treatment), and materials (composites, coatings), among many other fields.

  6. Rapidly growing market: From $200 billion in 2023 to a projected $380 billion in 2030 (CAGR of 9.8%). China and the United States lead the research.

  7. Safety and ethics: Risk assessment of biological and environmental effects is essential. Regulatory development, safe handling, and responsible research and innovation are required.

Nanomaterials are a fascinating field in which a single simple parameter—size—brings about dramatic changes in physical properties. Further development is expected in the future, spanning basic science to practical applications.

Preview of the Next Chapter

In the next chapter (Chapter 2: Fundamental Principles of Nanomaterials), we will study the following topics to gain a deeper understanding of the phenomena covered in this chapter:

The aim is to build a more quantitative understanding using mathematical formulas.

Exercises

Problem 1: Calculating Size Effects

For a spherical gold (Au) nanoparticle with a diameter of 10 nm, calculate the following.

(a) Find the surface-area-to-volume ratio.

(b) Taking the atomic radius of gold as 0.144 nm, estimate the approximate total number of atoms contained in this particle (Hint: the unit cell volume of gold's face-centered cubic lattice is about 0.068 nm³, with 4 atoms per unit cell).

(c) Estimate the approximate number of atoms present on the surface, and find the fraction of surface atoms (Hint: assume the surface atomic layer is a shell of 0.3 nm thickness).

Problem 2: Quantum Confinement Effect

Explain how the band gap changes when the size of a CdSe quantum dot decreases from 6 nm to 2 nm. Also describe the accompanying change in emission color. Provide a qualitative explanation of the change in energy levels from the perspective of the quantum confinement effect.

Problem 3: Classification of Nanomaterials

Classify the following nanomaterials as 0-dimensional, 1-dimensional, 2-dimensional, or 3-dimensional:

(a) Single-walled carbon nanotube (diameter 1 nm, length 10 μm)

(b) Graphene sheet (thickness 0.34 nm, lateral dimensions 1 mm × 1 mm)

(c) CdSe quantum dot (diameter 5 nm)

(d) MOF (metal-organic framework, crystal size 100 μm, pore size 1 nm)

(e) Silver nanowire (diameter 50 nm, length 20 μm)

(f) Fullerene C₆₀ (diameter approximately 0.7 nm)

Sample Answers ### Solution to Problem 1 **(a) Surface-area-to-volume ratio** Radius $r = 5$ nm = $5 \times 10^{-9}$ m $$ \frac{S}{V} = \frac{3}{r} = \frac{3}{5 \times 10^{-9}} = 6 \times 10^8 \text{ m}^{-1} $$ **Answer: $6 \times 10^8$ m⁻¹ = 600,000 m⁻¹** **(b) Total number of atoms** Volume of the particle: $$ V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (5 \times 10^{-9})^3 = 5.24 \times 10^{-25} \text{ m}^3 = 524 \text{ nm}^3 $$ Unit cell volume: $V\_{\text{cell}} = 0.068$ nm³, 4 atoms per unit cell $$ \text{Total number of atoms} = \frac{V}{V\_{\text{cell}}} \times 4 = \frac{524}{0.068} \times 4 \approx 30,800 \text{ atoms} $$ **Answer: approximately 30,000 atoms** **(c) Fraction of surface atoms** Volume of the surface shell (outer radius 5 nm, inner radius 4.7 nm): $$ V\_{\text{shell}} = \frac{4}{3}\pi (5^3 - 4.7^3) = \frac{4}{3}\pi (125 - 103.8) = 88.9 \text{ nm}^3 $$ Number of surface atoms: $$ N\_{\text{surface}} = \frac{88.9}{0.068} \times 4 \approx 5,230 \text{ atoms} $$ Fraction of surface atoms: $$ \frac{N\_{\text{surface}}}{N\_{\text{total}}} = \frac{5,230}{30,800} \approx 0.17 = 17\% $$ **Answer: approximately 5,200 surface atoms, a fraction of about 17%** (Note: a more rigorous calculation would take coordination number into account, but a simplified shell model is used here) --- ### Solution to Problem 2 **Change in band gap**: Due to the quantum confinement effect, the spacing between energy levels increases as particle size decreases. In the one-dimensional infinite well model: $$ E\_n \propto \frac{1}{L^2} $$ Therefore, when the size decreases from 6 nm to 2 nm (a factor of 1/3), the energy level spacing increases by about 9 times ($(1/3)^{-2} = 9$). Since this quantum confinement energy is added to CdSe's bulk band gap (1.74 eV): - **6 nm quantum dot**: band gap about 2.00 eV → **orange emission** (wavelength about 620 nm) - **2 nm quantum dot**: band gap about 2.75 eV → **blue emission** (wavelength about 450 nm) **Change in energy levels**: As particle size decreases, the motion of electrons and holes becomes more strongly confined, raising the ground-state energy. As a result, the maximum energy of the valence band decreases, the minimum energy of the conduction band increases, and the band gap consequently widens. **Change in emission color**: The emission shifts to shorter wavelengths, from orange → yellow-green → green → blue. --- ### Solution to Problem 3 | Nanomaterial | Classification | Reason | |---------|------|------| | (a) Single-walled CNT | **1-Dimensional** | Diameter 1 nm (nano-sized), length 10 μm (macro-sized) | | (b) Graphene sheet | **2-Dimensional** | Thickness 0.34 nm (nano-sized), lateral dimensions 1 mm (macro-sized) | | (c) CdSe quantum dot | **0-Dimensional** | All directions are 5 nm (nano-sized) | | (d) MOF | **3-Dimensional** | The overall crystal is 100 μm (macro), but contains 1 nm pores (nanostructure) internally | | (e) Silver nanowire | **1-Dimensional** | Diameter 50 nm (nano-sized), length 20 μm (macro-sized) | | (f) Fullerene C₆₀ | **0-Dimensional** | All directions are approximately 0.7 nm (nano-sized) |

References

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