MSE 201 S21 Lecture 37 - Module 1 - Free Energy of Nucleation

Summary

This module explores nucleation and the free energy involved in the process. It explains the roles of surface and bulk free energy during phase formation, with surface energy being destabilizing and volume energy being stabilizing. The total free energy of the system changes based on these interactions, leading to the concept of a critical nucleus. Once a nucleus reaches a critical radius (r*), it stabilizes and grows. The summary also highlights the formulas used to calculate the critical radius and energy, emphasizing how supercooling influences nucleation behavior.

Takeaways

• 🔬 Nucleation involves analyzing free energy, where negative values indicate favorability, and positive values imply that something is unlikely to occur.
• 🟥 The surface free energy is a positive term, representing the energy required to create a boundary between the solid and liquid phases. This destabilizes the nucleus and increases with particle radius.
• ⚖️ The surface free energy can be expressed as 4πr² multiplied by the surface tension between the interfaces.
• 🟩 The bulk free energy, a stabilizing factor, is related to the formation of a new solid phase and decreases as the particle radius increases.
• 📉 The bulk free energy term is proportional to the volume (r³) and decreases more rapidly than the surface term, which is proportional to r².
• 🌀 The total free energy of the system initially increases, reaching a maximum at a critical radius (r*), after which it decreases, indicating favorability for nucleation.
• 💡 The critical value (ΔG*) represents the energy required for nucleation to become stable, and once the radius exceeds r*, the nucleus will continue to grow.
• 📐 The critical radius (r*) is calculated using surface tension, melting temperature, latent heat of solidification, and the degree of supercooling (ΔT).
• ❄️ The critical nucleus size decreases as the supercooling increases, typically reaching about 10 nanometers for common supercooling values.
• ⚙️ The energy barrier for nucleation (ΔG*) also decreases with decreasing temperature, driven by the same variables that influence the critical radius.

Q & A

What is the basic concept of free energy in nucleation?

Free energy in nucleation helps determine whether a process is favorable or not. A negative free energy value indicates that the process is likely to occur, while a positive value means it is not likely to occur.

What are the two terms involved in the free energy of nucleation?

The two terms involved in the free energy of nucleation are the surface free energy (positive term) and the volume or bulk free energy (negative term). The surface free energy represents the energy required to create a surface between solid and liquid, while the volume free energy stabilizes the system by forming a new solid phase.

Why is the surface free energy term considered destabilizing?

The surface free energy term is considered destabilizing because forming a new boundary between the solid and liquid requires energy, making the nucleation process less favorable. This term increases with the radius of the particle.

How does the volume free energy term affect the nucleation process?

The volume free energy term is stabilizing and favorable for the system because it represents the energy behind forming a new solid phase. As the radius of the particle increases, the volume term decreases and becomes more negative, helping drive the nucleation process.

What happens when both surface and volume free energy terms are combined?

When the surface and volume free energy terms are combined, the total free energy initially increases but eventually starts to decrease once the volume term dominates. This leads to a critical value where the free energy becomes favorable for nucleation.

What is the significance of the critical radius (r*) in nucleation?

The critical radius (r*) is the size at which the nucleus becomes stable and will continue to grow. Before reaching this critical radius, the nucleus is just as likely to shrink as to grow. Once the radius exceeds r*, the nucleus becomes stable and can grow to further reduce the system's free energy.

What is delta G* in the context of nucleation?

Delta G* represents the critical amount of free energy that must be supplied to the nucleus for it to reach the stable size (r*). It is the energy barrier that must be overcome for nucleation to proceed.

How is the critical radius (r*) calculated?

The critical radius (r*) is calculated using the formula: r* = -2 * (surface tension or free energy) * (melting temperature) / (latent heat of solidification) * (degree of supercooling).

How does supercooling affect the critical radius and nucleation process?

As supercooling increases (i.e., the temperature decreases), the critical radius (r*) decreases. This means that smaller nuclei become stable at lower temperatures, making the nucleation process more likely.

What happens to the energy barrier for nucleation as the temperature decreases?

As the temperature decreases (with increasing supercooling), the energy barrier for nucleation (delta G*) decreases. This means it becomes easier for nuclei to form, which enhances the nucleation process.

Outlines

00:00

🧪 Understanding Free Energy in Nucleation

The first paragraph introduces the concept of free energy in nucleation, discussing how free energy helps determine whether a process is favorable or unfavorable. Negative free energy indicates a favorable process, while positive free energy means the opposite. The paragraph also highlights two components of nucleation: surface free energy (positive and unfavorable due to surface area formation) and volume free energy (favorable because the new phase formation decreases free energy). These components behave differently with changes in particle radius.

05:00

🔗 Surface Free Energy and Its Role in Nucleation

This section explains the surface free energy and its relation to the radius of a particle, showing that surface energy increases with radius due to the creation of new solid-liquid boundaries. It further explains that surface free energy is proportional to the square of the particle’s radius and is affected by the surface tension between interfaces. As the radius increases, surface energy grows, making nucleation less favorable until the bulk free energy takes over.

🔺 Volume Free Energy and Its Importance

The paragraph discusses the bulk or volume free energy, which is related to the formation of a new solid phase. As the radius of a particle increases, volume free energy becomes more negative and more favorable for nucleation. Since volume free energy decreases at a faster rate than surface free energy increases, a critical point is eventually reached where the total free energy starts to decrease, making nucleation more likely.

📉 Critical Point of Nucleation

This section describes the critical point in nucleation where the total free energy reaches its maximum before decreasing, allowing stable growth of nuclei. This critical point, denoted as 'delta G star,' is achieved when the radius of the particle reaches 'R star.' Below this size, the nuclei are as likely to shrink as they are to grow, but once the radius surpasses this critical value, the nuclei become stable and continue to grow.

⚖️ Calculating the Critical Radius

Here, the critical radius 'R star' is introduced as the radius beyond which nucleation becomes stable. The equation for 'R star' is derived using factors like surface tension, melting temperature, latent heat of solidification, and supercooling. It is explained that 'R star' decreases with increased supercooling, meaning that the critical nucleus size becomes smaller as the temperature drops.

🔢 Dependence of Critical Radius on Supercooling

This paragraph offers insight into how the critical radius, typically around 10 nanometers for typical supercooling degrees, changes with temperature. As supercooling increases (temperature decreases), the critical radius becomes smaller. The paragraph also notes that enthalpy and surface energy are weakly dependent on temperature, but these changes still influence the critical radius.

📊 Energy Barrier in Nucleation

This final section explains how the energy barrier for nucleation (delta G star) depends on surface free energy, melting temperature, latent heat of solidification, and supercooling. The energy required to form stable nuclei decreases as temperature decreases, further driving the process. The form of the equation changes slightly, but the variables remain consistent, allowing for a quantitative understanding of nucleation and how it relates to temperature.

Keywords

💡Nucleation

Nucleation refers to the initial process of phase transformation where small clusters (nuclei) of a new phase form in a material. In the video, nucleation is discussed in the context of the transition from liquid to solid during cooling. The video examines how free energy plays a role in determining whether nucleation is favorable or not.

💡Free Energy

Free energy is a thermodynamic concept that indicates the favorability of a process. Negative free energy means the process is likely to occur, while positive free energy suggests it is not. In the video, the free energy of nucleation is key to understanding the stability and formation of new phases during cooling.

💡Surface Free Energy

Surface free energy is the energy required to create a boundary between two phases, such as between a solid and liquid. This energy is unfavorable and contributes to the instability of a nucleus. The video explains that as the radius of the nucleus increases, surface free energy increases, making it harder for the phase transition to occur.

💡Volume Free Energy

Volume free energy refers to the energy change associated with the bulk of the new phase forming. In the case of solidification, this term is negative, meaning it favors the formation of the new phase. The video highlights that this decreases more rapidly than surface free energy as the size of the nucleus grows, eventually leading to a stable phase.

💡Critical Radius (r star)

The critical radius, denoted as 'r star', is the size of the nucleus at which it becomes stable and will continue to grow. The video shows that for nuclei smaller than the critical radius, it is just as likely to shrink as it is to grow. Beyond this point, nucleation is stable, and the system's free energy decreases.

💡Delta G Star (ΔG*)

Delta G Star represents the maximum free energy that needs to be supplied for a nucleus to become stable. It marks the energy barrier for nucleation. Once this energy threshold is reached, the nucleus can grow without further energy input. The video illustrates this concept by explaining that this energy barrier decreases as the system is further supercooled.

💡Supercooling

Supercooling refers to the process of cooling a liquid below its freezing point without it becoming a solid. The video uses this term to describe how the degree of supercooling affects the size of the critical radius and the free energy barrier, where increased supercooling leads to a smaller critical nucleus size.

💡Surface Tension

Surface tension is the force that acts at the interface between the liquid and solid phases. In the context of nucleation, it contributes to the surface free energy. The video discusses how surface tension plays a role in determining the energy required to form a new phase boundary.

💡Latent Heat of Solidification

Latent heat of solidification is the amount of heat released when a material changes from a liquid to a solid without changing temperature. This concept appears in the video as a factor in calculating the critical radius and the free energy associated with nucleation, affecting the phase transition dynamics.

💡Enthalpy

Enthalpy is the total heat content of a system and is used to describe the energy involved in phase transitions. The video mentions enthalpy in the context of calculating critical radius and free energy, explaining that while it is weakly temperature-dependent, it contributes to the overall energy balance during nucleation.

Highlights

Introduction to the concept of free energy and its role in determining the favorability of nucleation.

Negative free energy values indicate a likely occurrence, while positive values indicate an unlikely event.

Nucleation involves two key terms: surface free energy (unfavorable) and volume free energy (favorable).

Surface free energy relates to the formation of a new boundary between solid and liquid phases, which increases as particle radius increases.

Volume free energy decreases as the particle radius increases, contributing to the favorability of nucleation.

The combination of surface and volume free energies forms the total free energy, initially increasing but then decreasing when the volume term dominates.

The critical nucleus is reached at the maximum total free energy, denoted as delta G star and R star.

For radii smaller than R star, the nucleus is as likely to shrink as to grow, but for radii equal to or greater than R star, it stabilizes and grows.

Delta G star represents the energy required to stabilize a nucleus.

The critical radius (R star) can be calculated using surface tension, melting temperature, latent heat of solidification, and supercooling.

The critical radius decreases with increasing supercooling (decreasing temperature).

For typical degrees of supercooling, the critical radius is around 10 nanometers.

Delta G star also decreases with decreasing temperature, reducing the energy barrier for nucleation.

Surface free energy and enthalpy are weakly dependent on temperature, unlike the critical radius.

The transcript concludes by connecting the general ideas of nucleation and supercooling to specific calculations and principles.

Transcripts

all right in this module we're going to

continue talking about nucleation

and look at the free energy of

nucleation

all right so um again we're talking

about this concept of

free energy which you don't have to know

everything about this this is something

that you learn more about

in thermodynamics but

in a nutshell it basically tells us what

is

favorable or what is not favorable

so basically a negative value uh

means that something uh is likely to

occur

a positive it is not likely to occur

so that's the the basics of it um so

when we look at nucleation we're

actually going to look at

two different terms that happen when we

form

a new phase so the first one

is this positive term uh in red here the

red dashed

uh and this is related to the surface

free energy and what this is is it's

the um surface energy that's the surface

area that's created

between the solid and the liquid

interface and so

this takes energy this is not favorable

to form a new boundary

between solid and liquid and so that's

why this is positive it kind of de-state

what we say

is destabilizes the nuclei and so this

will

increase as the the radius

of the particle increases right because

the surface

energy will will decrease the surface

area will increase and therefore the

energy will increase

so that's what the red curve is showing

us and so this

can be related to uh the cir the

surface free energy can be related to

four pi

the radius squared of the particle

multiplied by

the surface tension between these

interfaces

the other part that is

stabilizing or favorable to the system

is what we call the volume or bulk free

energy

and so this is the free energy behind

the formation of this new solid

phase which again we know as favorable

based on

the fact that we're below the

equilibrium temperature

and so this is related to

the radius cubed because of volume right

so this is basically a volume term

multiplied by the

free energy per volume and so

in this case the term decreases

becomes more and more negative as the

radius increases

but as we know this is uh r

cubed and this is r squared so this is

going to do this

more rapidly than the surface term

so when we look at the combination of

both terms which gives us

the total free energy of the system

we're going to add those two terms

together right and so when we do that

because

this r cubed term decreases at a faster

rate than the r squared term

increases what we see is that

initially the total free energy

increases becomes less favorable but

at a certain point when this volume term

takes over

then we see it start to decrease and

eventually goes negative

and so we see here that this is a

maximum so this is kind of a critical

value

in this scenario because this is the

total amount of free energy

that we would have to put in the system

for this to

occur so basically what this is

showing us is that this is a critical

point so we call this

delta g star because what happens is

once the nuclei reaches the radius

here corresponding to this value

we call it r star so this is our star

because that's the maximum amount that

we have to put into the system

so um for radii less than our star

the nuclei is likely to sh is

just as likely to shrink as it is to

grow

however when the radius equals

that r star or greater then the nuclei

is said to be stable

and it will continue to grow to reduce

the free energy of the system so that's

why we call this kind of the critical

nucleus because that's where it becomes

stable below that it's also likely to

go back to uh to liquid

so what we have to do is we basically

have to supply delta g

star's worth of energy to the nuclei

before it becomes stable so that's why

we call it a critical

value and so

if we take the terms for volume and

surface and

add them together we saw that we had

that and we can actually

solve for this maximum and solve for our

star

by by looking for the maximum of this

equation

and if we solve for that we see that the

critical radius r star

is equal to minus two times the surface

tension or surface free energy

multiplied by the melting temperature

divided by the latent heat of

solidification so the enthalpy

multiplied by the supercooling so delta

t

and so this allows us to calculate the

r star for this system

and what we see is that it's going to be

a function

uh inverse function of the

the delta t the super cooling term

um a couple other notes to mention here

so enthalpy

and surface energy are weakly dependent

on this temperature but not as much

as the r star so it's just a couple

these can change with temperature

but um for the most part they're only

weakly uh dependent

so what we see is that our star will

decrease

as supercooling increases right so the

temperature

decreases overall so

this size of the critical nuclei becomes

smaller

with decreasing temperature or

increasing delta t

and so if you're wondering some some

approximate values here what these

values tend to be

so for typical degrees of supercooling

the critical radius is about 10

nanometers

right so that's when it becomes critical

so about 10 nanometers

all right so along with that we can

calculate the delta

g star and that's the when

the we reach the critical radius right

the energy value there

and this has a similar dependence right

it's just um

the form of it's different but it has

the same components we have the

gamma which is the surface free energy

we have melting temperature

the latent heat of solidification delta

h f

and then the degree of supercooling and

that's squared

so different form but same variables

in this case and what we see is that the

barrier

for forming a nuclei decreases

with increase with decreasing

temperature right that's what we saw

that's what we kind of

said in general happens um the direct

because this is again the driving force

but this will

allows us to put some numbers behind

the these general ideas