Nucleation and Growth Kinetics

Introduction to Materials Science and Engineering 21089201

Chedtha Puncreobutr

Department of Metallurgical Engineering Chulalongkorn University

Nucleation and Growth Kinetics

Phase transformation

Nucleation - The physical process by which a new phase is produced in a material. It is the initial process in crystallization.

Most phase transformations begin with the formation of numerous small particles of the new phase that increase in size until the transformation is

complete.

Nucleation

• Nucleation is the process of forming a nucleus

• It is the process in which ions, atoms, or molecules arrange themselves in a pattern characteristic of a crystalline solid, forming a site in which additional particles deposit as the crystal grows

• Some examples of phases that may form by way of nucleation in liquids are gaseous bubbles, crystals, or glassy regions.

• Creation of liquid droplets in saturated vapour is also characterized by nucleation

Nucleation mechanisms

• Homogeneous nucleation

Formation of a critically sized solid from the liquid by the clustering together of a large number of atoms at a high undercooling (without an external interface).

• Heterogeneous nucleation

Formation of a critically sized solid from the liquid on

an impurity surface.

Driving force

The change in free energy DG (function of the internal energy and enthalpy of the system) must be negative for a transformation to occur.

𝑻 > 𝑻_𝒎 𝑻 < 𝑻_𝒎

liquid solid

∆𝐺 = ∆𝐻 − 𝑇∆𝑆

∆𝐺 = 0 = ∆𝐻 − 𝑇

_𝑚

∆𝑆

𝑎𝑡 𝑇_𝑚 Undercooling:

∆𝑆 = ∆𝐻

𝑇

_𝑚

= ∆𝐻

_𝑓

𝑇

_𝑚 entropy of fusion Temperature

Molar free energy (DG)

𝐺_𝑆 𝐺_𝐿

∆𝐺

𝑇_𝑚 𝑇

∆𝑇

for small∆𝑇

∆𝐺 = ∆𝐻

_𝑓

− 𝑇 ∆𝐻

_𝑓

𝑇

_𝑚

∆𝐺 ≅ ∆𝐻

_𝑓

𝑇

_𝑚

∆𝑇

∆𝑇 = 𝑇_𝑚 − 𝑇

Homogeneous Nucleation

• Assume that nuclei of the solid phase form spontaneously in the interior of the liquid as atoms cluster together- similar to the packing in the solid phase.

• Also, each nucleus is spherical and has a radius r

• Free energy changes as a result of a transformation:

1) the difference between the solid and liquid phases (volume free energy, DG

_V

)

2) the solid-liquid phase boundary

(surface free energy, DG

_S

). ∆𝐺 = −𝑉

_𝑠

∆𝐺

_𝑣

+ 𝐴

_𝑠𝑙

𝛾

_𝑠𝑙

Homogeneous Nucleation & Energy Effects

DG

_T

= Total Free Energy

= DG

_S

+ DG

_V

Surface Free Energy - destabilizes the nuclei (it takes energy to make an interface)







D G

_S

4 r

²

 = surface tension

Volume (Bulk) Free Energy –

stabilizes the nuclei (releases energy)

D







D G

_V

r

³

G 3

4

∆𝐺

_𝑉

= − 4

3 𝜋𝑟

³

∆𝐻

_𝑓

𝑇

_𝑚

∆𝑇

Homogeneous Nucleation & Critical radius

∆𝐺 = −4

3𝜋𝑟³∆𝐻_𝑓

𝑇_𝑚 ∆𝑇 + 4𝜋𝑟²𝛾_𝑆𝐿

𝑑∆𝐺

𝑑𝑟 = 0 at 𝑟^∗

∆𝐺^∗ = 16𝜋𝛾_𝑆𝐿³𝑇_𝑚 ² 3∆𝐻_𝑓²∆𝑇² 𝑟 < 𝑟^∗

𝑟 > 𝑟^∗

embryos - melt back nuclei - grow

(∆𝐺_{ℎ𝑜𝑚𝑜})

𝑟^∗ = 2𝛾_𝑆𝐿𝑇_𝑚

∆𝐻_𝑓∆𝑇 surface free energy α 𝑟²

volume free energy α 𝑟³∆𝑇

Homogeneous Nucleation & Critical radius

∆𝐺^∗ = 16𝜋𝛾_𝑆𝐿³𝑇_𝑚 ² 3∆𝐻_𝑓²∆𝑇² 𝑟^∗ = 2𝛾_𝑆𝐿𝑇_𝑚

∆𝐻_𝑓∆𝑇  r* decreases as DT increases For typical DT r* ~ 10 nm

Critical Radius for the Solidification of Copper

Calculate the size of the critical radius and the number of atoms in the critical

nucleus when solid copper forms by homogeneous nucleation.

Rate of formation of homogeneous nuclei

r

Number of stable nuclei 𝑛^∗ = 𝑛_𝐿𝑒𝑥𝑝 −∆𝐺^∗

𝑘_𝐵𝑇

= Boltzmann’s constant 𝑘_𝐵

𝑣_𝑑 atomic vibration frequency

𝑝_𝑐 probability of capturing an atom at the surface

Rate of formation of homogeneous nuclei

ሶ 𝑁

_ℎ𝑜𝑚

= 𝑣

_𝑑

𝑝

_𝑐

𝑛

_𝐿

𝑒𝑥𝑝 − 16𝜋 3

𝛾

_𝑆𝐿³

𝑇

_𝑚²

𝑘

_𝐵

𝑇∆𝐻

_𝑓²

∆𝑇

²

• At high temperatures, the thermodynamic driving force for nucleation becomes less due to the lower undercooling and the number of critical clusters will be limited.

• At low temperatures, the thermodynamic driving force for the nucleation is large due to the large undercooling, but the rate of diffusion of atoms to the nucleation site is reduced

Heterogeneous nucleation

Heterogeneous nucleation

When solidification is initiated from a foreign surface, i.e. solid particles suspended in liquid, oxide layers, or surface contact with a crucible wall, it is said to nucleate heterogeneously

Assuming a spherical cap shape of nucleus formed on a surface of foreign substrate

Heterogeneous nucleation

The positive interfacial energy term can be reduced by a factor

∆𝐺

_{ℎ𝑒𝑡𝑒𝑟}

= 16𝜋𝛾

_𝑆𝐿³

𝑇

_𝑚 ²

3∆𝐻

_𝑓²

∆𝑇

²

𝑓(𝜃)

∆𝐺

_{ℎ𝑒𝑡𝑒𝑟}

= ∆𝐺

_{ℎ𝑜𝑚𝑜}

∙ 𝑓 𝜃

𝑓(𝜃)

 = wetting angle

𝑓(𝜃) geometric factor given by ratio of the

volumes of spherical cap and a full sphere of identical radius

Since the heterogeneous nucleation barrier is lower than that of homogeneous

nucleation, it is much easier for heterogeneous nucleation to occur during solidification.

Heterogeneous nucleation

∆𝐺_{ℎ𝑒𝑡𝑒𝑟} = 16𝜋𝛾_𝑆𝐿³𝑇_𝑚²

3∆𝐻_𝑓²∆𝑇² 𝑓(𝜃)

= 16𝜋𝛾_𝑆𝐿³𝑇_𝑚 ² 3∆𝐻_𝑓²∆𝑇²

Heterogeneous nucleation

• Lower free energy for heterogeneous nucleation means a smaller energy to overcome during nucleation process, therefore, heterogeneous nucleation occurs more readily

• Much smaller degree of undercooling is required for heterogeneous

nucleation.

Growth

ሶ 𝐺 = 𝐶 𝑒𝑥𝑝 − 𝑄 𝑘

_𝐵

𝑇

• Growth rate is determined by the rate of diffusion, and its temperature dependence is the same as for the

diffusion coefficient

• Growth step in a phase transformation begins once an embryo has exceeded the critical size and becomes a stable nucleus

• Particle growth occurs by long-range atomic diffusion

Growth

• Transformation occurs near melting point, low nucleation but high growth rate. Thus, resulting microstructure will consist few and relatively large particles (e.g. coarse grains)

• Nucleation will continue to occur simultaneously with growth of the new phase particles. At a specific temperature, the overall transformation rate is equal to some product of nucleation and growth rates

• The size of the product phase particles will depend on transformation temperature

• Conversely, for transformations at lower temperatures, nucleation rates are high and growth rates low, which results in many small particles (e.g., fine grains).