Metal Nanocrystals by Reduction 

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1. Introduction

Metal nanocrystals are synthesized by reducing metal ions ($M^{n+}$) to zero-valent metal atoms ($M^0$) using a chemical reducing agent, followed by nucleation and growth to form nanoscale crystalline particles.

Core Reaction: $M^{n+}+n{e}^{-}\stackrel{\text{reducing agent}}{\to }{M}^0$

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2. Thermodynamics of Nucleation

2.1 Classical Nucleation Theory (CNT)

The Gibbs free energy change for forming a spherical nucleus of radius $r$:

$$\mathrm{\Delta }G=\mathrm{\Delta }{G}_{volume}+\mathrm{\Delta }{G}_{surface}$$

$$\mathrm{\Delta }G=-\frac{4}{3}\pi r^3\cdot \frac{RT\ln S}{V_m}+4\pi r^2\gamma$$

Where:

• $S=\frac{[M^{n+}]}{[M^{n+}{]}_{eq}}$ = supersaturation ratio
• $V_m$ = molar volume of the metal
• $\gamma$ = surface free energy (interfacial tension)

2.2 Critical Nucleus Radius ($r_c$)

At the critical point, $\frac{d\mathrm{\Delta }G}{dr}=0$:

$$r_c=\frac{2\gamma V_m}{RT\ln S}$$

Key insight: Higher supersaturation → smaller critical radius → easier nucleation

2.3 Activation Energy for Nucleation ($\mathrm{\Delta }{G}_c$)

$$\mathrm{\Delta }{G}_c=\frac{16\pi {\gamma }^3{V}_m^2}{3(RT\ln S{)}^2}$$

Key insight: Higher $\gamma$ or lower $S$ → higher energy barrier → harder to nucleate

2.4 Nucleation Rate ($J$)

$$J=A\cdot \exp (-\frac{\mathrm{\Delta }{G}_c}{k_{B}T})\cdot \exp (-\frac{E_a}{k_{B}T})$$

Where:

• $A$ = pre-exponential factor
• $E_a$ = activation energy for atomic attachment

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3. Nucleation Mechanisms

3.1 Burst Nucleation (LaMer Mechanism)

Three stages:

less
Concentration
    ↑
    │        Stage I:         Stage II:         Stage III:
    │    No nucleation    Burst nucleation    Growth by diffusion
    │                        │
C_max ──────────────────────►│◄────────────────────
    │                        │
C_min ──────────────────────►│◄────────────────────
    │                        │
C_eq ────────────────────────┴─────────────────────► Time

• Stage I: Metal ion concentration increases (no nucleation)
• Stage II: At $C_{max}$ (critical supersaturation), burst nucleation occurs → concentration drops
• Stage III: Below $C_{min}$, no new nuclei form; existing nuclei grow by diffusion

LaMer's key insight: Separation of nucleation and growth is essential for monodisperse nanocrystals.

3.2 Finke-Watzky Mechanism (Slow Continuous Nucleation)

• Nucleation is slow and continuous throughout the reaction
• Growth occurs on existing nuclei (autocatalytic surface growth)
• Produces broader size distributions

$$Rate=k_1[M^{n+}]+k_2[M^{n+}][M_{surface}^0]$$

Where $k_1$ = nucleation rate, $k_2$ = autocatalytic growth rate

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4. Growth Mechanisms

4.1 Diffusion-Limited Growth

When diffusion of metal atoms to the surface is the slowest step:

$$\frac{dr}{dt}=\frac{D{V}_m(C_b-C_s)}{r}$$

Where:

• $D$ = diffusion coefficient
• $C_b$ = bulk concentration
• $C_s$ = surface concentration

4.2 Reaction-Limited Growth

When surface reaction is the slowest step:

$$\frac{dr}{dt}=k_r{V}_m(C_s-C_{eq})$$

Where $k_r$ = surface reaction rate constant

4.3 Ostwald Ripening

Large particles grow at the expense of small particles due to higher surface energy of small particles:

Gibbs-Thomson relation:

$$C(r)=C_{eq}\cdot \exp \left(\frac{2\gamma V_m}{rRT}\right)$$

Smaller $r$ → higher $C(r)$ → smaller particles dissolve → larger particles grow