This document summarizes phase-field modeling of homogeneous crystal nucleation using two main methods. The first method adds fluctuations (noise) to the phase-field equations of motion to mimic natural nucleation. The noise amplitude is determined by the fluctuation-dissipation theorem. This models nucleation without assuming a sharp interface or bulk properties. The second method places supercritical crystal seeds randomly in space and time to model nucleation. Quantitative results from both methods are difficult to obtain due to limitations of classical nucleation theory. The document outlines the phase-field model and equations used, including discretization for simulation. It demonstrates convergence of results with decreasing grid spacing and time step when modeling crystal growth and nucleation with noise.
Localized Electrons with Wien2k
LDA+U, EECE, MLWF, DMFT
Elias Assmann
Vienna University of Technology, Institute for Solid State Physics
WIEN2013@PSU, Aug 14
Kinetic pathways to the isotropic-nematic phase transformation: a mean field ...Amit Bhattacharjee
Here we illustrate the classic Ginzburg-Landau-de Gennes theory of isotropic nematic phase transition and show how fluctuations as well as deterministic kinetics can lead to phase equilibria.
A seminar presented in "CompFlu16" at IIIT Hyderabad in December 2016 on homogeneous nucleation kinetics in anisotropic liquids using a Landau-de Gennes field theoretic study.
This talk was presented at the 22nd International conference on Surface Modification Technology, 22-24 September 2008, in Trollhattan, Sweden. It describes some recent computational research work carried out using molecular dynamics methods to calculate physical properties, including viscosity, of liquid nickel over a wide temperature range.
This is the plenary talk given by Prof Shyue Ping Ong at the 57th Sanibel Symposium held on St Simon's Island in Georgia, USA.
Abstract: Powered by methodological breakthroughs and computing advances, electronic structure methods have today become an indispensable toolkit in the materials designer’s arsenal. In this talk, I will discuss two emerging trends that holds the promise to continue to push the envelope in computational design of materials. The first trend is the development of robust software and data frameworks for the automatic generation, storage and analysis of materials data sets. The second is the advent of reliable central materials data repositories, such as the Materials Project, which provides the research community with efficient access to large quantities of property information that can be mined for trends or new materials. I will show how we have leveraged on these new tools to accelerate discovery and design in energy and structural materials as well as our efforts in contributing back to the community through further tool or data development. I will also provide my perspective on future challenges in high-throughput computational materials design.
On prognozisys of manufacturing double basemsejjournal
In this paper we introduce a modification of recently introduced analytical approach to model mass- and
heat transport. The approach gives us possibility to model the transport in multilayer structures with account
nonlinearity of the process and time-varing coefficients and without matching the solutions at the
interfaces of the multilayer structures. As an example of using of the approach we consider technological
process to manufacture more compact double base heterobipolar transistor. The technological approach
based on manufacturing a heterostructure with required configuration, doping of required areas of this heterostructure
by diffusion or ion implantation and optimal annealing of dopant and/or radiation defects. The
approach gives us possibility to manufacture p-n- junctions with higher sharpness framework the transistor.
In this situation we have a possibility to obtain smaller switching time of p-n- junctions and higher compactness
of the considered bipolar transistor.
Tem for incommensurately modulated materialsJoke Hadermann
This presentation is a teaching lecture given on the International School on Aperiodic Crystals and explains how to index electron diffraction patterns taken from incommensurately modulated materials, with exercises, and gives some examples of HAADF-STEM and HRTEM images on incommensurately modulated materials.
On prognozisys of manufacturing doublebaseijaceeejournal
In this paper we introduce a modification of recently introduced analytical approach to model mass- and
heat transport. The approach gives us possibility to model the transport in multilayer structures with account
nonlinearity of the process and time-varing coefficients and without matching the solutions at the
interfaces of the multilayer structures. As an example of using of the approach we consider technological
process to manufacture more compact double base heterobipolar transistor. The technological approach
based on manufacturing a heterostructure with required configuration, doping of required areas of this
heterostructure by diffusion or ion implantation and optimal annealing of dopant and/or radiation defects.
The approach gives us possibility to manufacture p-n- junctions with higher sharpness framework the transistor.
In this situation we have a possibility to obtain smaller switching time of p-n- junctions and higher
compactness of the considered bipolar transistor.
Localized Electrons with Wien2k
LDA+U, EECE, MLWF, DMFT
Elias Assmann
Vienna University of Technology, Institute for Solid State Physics
WIEN2013@PSU, Aug 14
Kinetic pathways to the isotropic-nematic phase transformation: a mean field ...Amit Bhattacharjee
Here we illustrate the classic Ginzburg-Landau-de Gennes theory of isotropic nematic phase transition and show how fluctuations as well as deterministic kinetics can lead to phase equilibria.
A seminar presented in "CompFlu16" at IIIT Hyderabad in December 2016 on homogeneous nucleation kinetics in anisotropic liquids using a Landau-de Gennes field theoretic study.
This talk was presented at the 22nd International conference on Surface Modification Technology, 22-24 September 2008, in Trollhattan, Sweden. It describes some recent computational research work carried out using molecular dynamics methods to calculate physical properties, including viscosity, of liquid nickel over a wide temperature range.
This is the plenary talk given by Prof Shyue Ping Ong at the 57th Sanibel Symposium held on St Simon's Island in Georgia, USA.
Abstract: Powered by methodological breakthroughs and computing advances, electronic structure methods have today become an indispensable toolkit in the materials designer’s arsenal. In this talk, I will discuss two emerging trends that holds the promise to continue to push the envelope in computational design of materials. The first trend is the development of robust software and data frameworks for the automatic generation, storage and analysis of materials data sets. The second is the advent of reliable central materials data repositories, such as the Materials Project, which provides the research community with efficient access to large quantities of property information that can be mined for trends or new materials. I will show how we have leveraged on these new tools to accelerate discovery and design in energy and structural materials as well as our efforts in contributing back to the community through further tool or data development. I will also provide my perspective on future challenges in high-throughput computational materials design.
On prognozisys of manufacturing double basemsejjournal
In this paper we introduce a modification of recently introduced analytical approach to model mass- and
heat transport. The approach gives us possibility to model the transport in multilayer structures with account
nonlinearity of the process and time-varing coefficients and without matching the solutions at the
interfaces of the multilayer structures. As an example of using of the approach we consider technological
process to manufacture more compact double base heterobipolar transistor. The technological approach
based on manufacturing a heterostructure with required configuration, doping of required areas of this heterostructure
by diffusion or ion implantation and optimal annealing of dopant and/or radiation defects. The
approach gives us possibility to manufacture p-n- junctions with higher sharpness framework the transistor.
In this situation we have a possibility to obtain smaller switching time of p-n- junctions and higher compactness
of the considered bipolar transistor.
Tem for incommensurately modulated materialsJoke Hadermann
This presentation is a teaching lecture given on the International School on Aperiodic Crystals and explains how to index electron diffraction patterns taken from incommensurately modulated materials, with exercises, and gives some examples of HAADF-STEM and HRTEM images on incommensurately modulated materials.
On prognozisys of manufacturing doublebaseijaceeejournal
In this paper we introduce a modification of recently introduced analytical approach to model mass- and
heat transport. The approach gives us possibility to model the transport in multilayer structures with account
nonlinearity of the process and time-varing coefficients and without matching the solutions at the
interfaces of the multilayer structures. As an example of using of the approach we consider technological
process to manufacture more compact double base heterobipolar transistor. The technological approach
based on manufacturing a heterostructure with required configuration, doping of required areas of this
heterostructure by diffusion or ion implantation and optimal annealing of dopant and/or radiation defects.
The approach gives us possibility to manufacture p-n- junctions with higher sharpness framework the transistor.
In this situation we have a possibility to obtain smaller switching time of p-n- junctions and higher
compactness of the considered bipolar transistor.
Major Goal: estimate risks of the pollution in a subsurface flow.
How? We solve density-driven groundwater flow with uncertain porosity and permeability.
1. We set up density-driven groundwater flow problem
2. Review stochastic modeling and stochastic methods
3. Modeling of uncertainty in porosity and permeability
4. Numerical methods to solve deterministic problem
5. 2D and 3D examples with 0.5-8 Millions mesh points.
Mobility Measurements Probe Conformational Changes in Membrane-embedded prote...richardgmorris
The function of membrane-embedded proteins such as ion channels depends crucially on their conformation. We demonstrate how conformational changes in asymmetric membrane proteins may be inferred from measurements of their diffusion. Such proteins cause local deformations in the membrane, which induce an extra hydrodynamic drag on the protein. Using membrane tension to control the magnitude of the deformations and hence the drag, measurements of diffusivity can be used to infer--- via an elastic model of the protein--- how conformation is changed by tension. Motivated by recent experimental results [Quemeneur et al., Proc. Natl. Acad. Sci. USA, 111 5083 (2014)] we focus on KvAP, a voltage-gated potassium channel. The conformation of KvAP is found to change considerably due to tension, with its `walls', where the protein meets the membrane, undergoing significant angular strains. The torsional stiffness is determined to be 26.8 kT at room temperature. This has implications for both the structure and function of such proteins in the environment of a tension-bearing membrane.
Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...Alexander Litvinenko
1. Solved time-dependent density driven flow problem with uncertain porosity and permeability in 2D and 3D
2. Computed propagation of uncertainties in porosity into the mass fraction.
3. Computed the mean, variance, exceedance probabilities, quantiles, risks.
4. Such QoIs as the number of fingers, their size, shape, propagation time can be unstable
5. For moderate perturbations, our gPCE surrogate results are similar to qMC results.
6. Used highly scalable solver on up to 800 computing nodes,
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
The Phase Field Method: Mesoscale Simulation Aiding Materials DiscoveryPFHub PFHub
Two types of computational materials science, model development and materials discovery. PF is used less than atomic scale methods. PF focused on model development not discovery. How to use PF for materials discovery?
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Phase-field modeling of crystal nucleation I: Fundamentals and methods
1. Tamás Pusztai
Wigner Research Centre for Physics
Hungarian Academy of Sciences
Budapest, Hungary
Phase Field Workshop
ChiMaD Headquarters, Northwestern University
Sept 25-27, 2018
Phase-field modeling of crystal nucleation I:
Fundamentals and methods
2. Introduction
Solidification of an undercooled liquid
GrowthNucleation
200,000 time steps 220,000 time steps
160,000 time steps 180,000 time steps
Molecular dynamics simulation: By courtesy of R. S-Aga & J. R. Morris
Embryos of the new phase appear
via thermal fluctuation
Complex patterns evolve due to the
interplay of capillarity, diffusion,
and anisotropy.
3. Outline
• Homogeneous nucleation
• theory
• modeling
• problems
• Heterogeneous nucleation
• theory
• modeling of surfaces with adjustable wetting properties
• Greer’s athermal nucleation model
• theory
• modeling
4. Homogeneous nucleation
Classical nucleation theory (CNT):
assumptions:
• sharp interface
• bulk properties inside
• spherical shape (isotropic surface energy)
F3D(r) = 4r2
⇡ SL
4
3
r3
⇡ f F2D(r) = 2r⇡ SL r2
⇡ f
r⇤
3D =
2 SL
f
r⇤
2D =
SL
f
,
W⇤
3D =
16⇡ 3
SL
3 f2
W⇤
2D =
⇡ 2
SL
f
✓
Gibbs-Thomson: f = 2 sl, =
1
2
✓
1
R1
+
1
R2
◆
, 3D =
1
r
2D =
1
2r
◆
propertyofTamásPusztai(pusztai.tamas@wigner.mta.hu)-August22,201313:24
Homogeneous nucleation
4 × 10––15
2 × 10––15
––2 × 10––15
––4 × 10––15
1 × 10––8 2 × 10––8 3 × 10––8
R [m]
Volume term
Area term
DΔG(R)
Freeenergy[J]
4 × 10––8 5 × 10––8 6 × 10––8
0
0
DΔGn
homo
Rc
Fig. 7.1 Surface, bulk and total free energies of a spherical solid as function
radius for a fixed undercooling T = 5 K. Property data for Al are tabula
Table 7.1.
this occurs is determined by differentiating Eq. (7.2) with respect to
setting the result equal to zero:
Rc =
2 s`
⇢ sf T
=
2 s`
T
where s` = s`/(⇢s sf ) is the Gibbs-Thomson coefficient. Substi
this result into Eq. (7.2) gives
Ghomo
n =
4⇡ s`R2
c
3
=
16⇡
3
3
s`
(⇢ sf )2 T2
An embryo of radius Rc is called a critical nucleus, since it is energe
favorable for nuclei with R < Rc to melt, and for R > Rc to grow. Not
for T < 0, corresponding to temperatures above Tf , both terms o
Free energy of a spherical solid particle
of radius r: M. Rappaz: Solidification
2D3D
5. Homogeneous nucleation
J = J0 exp
✓
W⇤
kT
◆
Nucleation rate:
• Nucleation rates corresponding to time scales of typical experiments may
correspond to 10-100 molecules → the CNT fails miserably → a diffuse interface
model is needed!
• The nucleus is not necessarily spherical
extremely sensitive to W!
Limits of the classical theory:
milar bond-order pa-
ore said to be joined
” (26). Even in the
al-like bonds are not
particles with eight
onds are defined as
particles in a crys-
, hcp, rhcp, or bcc
recognized, whereas
umber of particles in
be crystal-like.
an experiment, sam-
astable liquid state,
m structural fluctua-
i of crystal-like par-
ticles were present. This is shown in two
early-time snapshots (Fig. 1, A and B),
where we represent crystal-like particles as
red spheres and liquid-like particles as blue
spheres, shown with a reduced diameter to
improve visibility. Typically, these sub-
critical nuclei contained no more than 20
particles and shrank to reduce their surface
energy. After a strongly -dependent peri-
od of time, critical nuclei formed and rap-
idly grew into large postcritical crystallites
(Fig. 1, C and D). By following the time
evolution of many crystallites, we deter-
mined the size dependence of the probabil-
ities pg and ps with which crystallites grow
or shrink (27). Because pg ϭ ps at the
critical size, we plot the difference pg – ps
as a function of crystallite radius and par-
ticle number M in Fig. 2 for a sample with
ϭ 0.47. We found an abrupt change from
negative to positive values of pg – ps (28),
allowing us to identify the critical size,
which is 60 Ͻ M Ͻ 160, in good agreement
with recent computer simulations (9). This
corresponds to rc Ϸ 6.2a, assuming a spher-
ical nucleus. The volume fraction of the
nuclei is larger than the value of the
fluid; above coexistence, the difference is
⌬ ϭ 0.012 Ϯ 0.003, independent of ,
where ⌬ increases slightly for M Ͼ 100.
We can understand this ⌬ value as result-
ing from the higher osmotic pressure exert-
ed by the fluid on the nuclei (16), whereas
in the coexistence regime, ⌬ must reflect
the evolution of to the higher value,
ultimately attained by the crystallites,
where ⌬ ϭ m Ϫ f. The nucleation rate
densities were slower than 5 mmϪ3
sϪ1
for
Ͻ 0.45, as well as for Ͼ 0.53. Values
of the order of 10 mmϪ3
sϪ1
were found for
0.45 Ͻ Ͻ 0.53. However, for 0.47 Ͻ Ͻ
0.53, the average size of the nuclei began to
grow immediately after shear melting; thus,
there was little time for the sample to equil-
a
l
ϭ
e
A
d
-
e
3
e
e
e
s
-
R E P O R T S
disordered liquid, crystal-like bonds are not
uncommon; thus, only particles with eight
or more crystal-like bonds are defined as
being crystal-like. All particles in a crys-
tallite with perfect fcc, hcp, rhcp, or bcc
structure are correctly recognized, whereas
only an insignificant number of particles in
the liquid are found to be crystal-like.
At the beginning of an experiment, sam-
ples started in the metastable liquid state,
but because of random structural fluctua-
tions, subcritical nuclei of crystal-like par-
red spheres and liquid-like particles as blue
spheres, shown with a reduced diameter to
improve visibility. Typically, these sub-
critical nuclei contained no more than 20
particles and shrank to reduce their surface
energy. After a strongly -dependent peri-
od of time, critical nuclei formed and rap-
idly grew into large postcritical crystallites
(Fig. 1, C and D). By following the time
evolution of many crystallites, we deter-
mined the size dependence of the probabil-
ities pg and ps with which crystallites grow
Fig. 3. A snapshot of a
crystallite of postcritical
size in a sample with ϭ
0.47 is shown from three
different directions (A
through C). The 206 red
spheres represent crystal-
like particles and are
drawn to scale; the 243
extra blue particles share
at least one crystal-like
“bond” to a red particle
but are not identified as
crystal-like and are re-
duced in size for clarity.
(D) A cut with a thickness
of three particle layers
through the crystallite, il-
lustrating the hexagonal
structure of the layers.
Blue, red, and green
spheres represent parti-
cles in the different layers
(front to rear). This cut
was taken from the re-
gion that is indicated by
Gasser et al, Science, 2001
6. Phase-field modeling of homogeneous nucleation
• Method 1: Mimic nature. Add fluctuations (noise) and wait
Determine W* by finding the nucleus (saddle point solution) via solving the Euler-
Lagrange equations.
Sharp interface and bulk properties are not assumed any more
∂ϕ
∂t
= − M
δF
δϕ
+ξ
∂ϕ
∂c
= ∇
[
M ∇
δF
δc
+ ⃗ζ
]
• Method 2: Put supercritical seeds at random places at random times
J = J0 exp
(
−
W*
kT )
Nucleation rate:
Remark: quantitative results are difficult, to say the least…
where ξ and ⃗ζ are gaussian random variables with
zero mean that put the “right amount” of fluctuations into the system
7. Method 1: Homogeneous nucleation with noise
∂ϕ
∂t
= − M
δF
δϕ
+ξ( ⃗r, t) with
With the noise term added, the equation of motion becomes a stochastic PDE, the noise
amplitude is determined by the fluctuation-dissipation theorem
Hohenberg-Halperin classification:
(P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena. Reviews of Modern Physics, 49, 435–479, 1977)
Model A (Non-conserved field):
Model C (Non-conserved field coupled to a conserved field):
∂ϕ
∂t
= − Mϕ
δF
δϕ
+ξ( ⃗r, t) with < ξ( ⃗r, t)ξ( ⃗r′, t′) > = 2MϕkT δ( ⃗r − ⃗r′)δ(t − t′)
∂c
∂t
= ∇
[
Mc ∇
δF
δc
+ ⃗ζ ( ⃗r, t)
]
with < ζm( ⃗r, t)ζn( ⃗r′, t′) > = 2MckT δm,nδ( ⃗r − ⃗r′)δ(t − t′)
< ξ( ⃗r, t) > = 0
< ξ( ⃗r, t)ξ( ⃗r′, t′) > = 2MkT δ( ⃗r − ⃗r′)δ(t − t′)
8. Method 1: Homogeneous nucleation with noise
Discretization:
ϕt+1
x = ϕt
x + Δt M
(
ϵ2
ϕt
x+1 + ϕt
x−1 − 2ϕt
x
Δx2
− g′(ϕt
x) + p′(ϕt
x)
)
+ Δt
MkT
ΔxdΔt
ξ
In 2D or 1D simulation we still use 3D materials parameters → some thickness is
always implicitly assumed.
This is not a problem in the deterministic part, but results in a different scaling of the
stochastic part!
< ξ(r) ξ( ⃗r′) > = 2MkT δ( ⃗r − ⃗r′) =
2MkT
ΔxdΔt
δn,n′, where d = number of spatial dimensions
Complete equation of motion with finite difference and forward Euler:
Mind the dimensionality!!!
< ξ > = 0
< ξ2
> = 1
deterministic part stochastic part
9. The 2D “toy model” used for illustrations
F[ϕ] =
∫ [
ϵ2
2
(∇ϕ)2
+ wg(ϕ) − Δf(T) p(ϕ)
]
dV
g(ϕ) = ϕ2
(1 − ϕ)2
p(ϕ) = ϕ3
(10 − 15ϕ + 6ϕ2
)
Simple phase-field model for a pure substance
Equilibrium solid-liquid interface:
ϕ(x) =
1 − tanh
(
x
2δ )
2
δ =
ϵ2
w
, γ =
ϵ2
w
3 2
Nucleation and growth in 2D:
T = Tm, Δf = 0
T < Tm, Δf > 0
r*2D
=
γ
Δf
=
ϵ2
w
3 2Δf
·
ϕ = − M
δF
δϕ
= M (ϵ2
∇2
ϕ − wg′(ϕ) + Δf p′(ϕ)) +
MkT
Δx2Δt
ξ
ϕ′(x) =
2w
ϵ2
ϕ(1 − ϕ)
10. 0 20 40 60 80
time
0
0.2
0.4
0.6
0.8
1
solidfraction
t 0
t 0
/2
t 0
/4
t 0
/8
t 0
/16
0 20 40 60 80
time
0
0.2
0.4
0.6
0.8
1
solidfraction
x 0
x 0
/2
x 0
/4
Solid seed growing without noise, convergence
Δx0 = 0.4, Δt0 = 0.01 Δx0 = 0.4, Δt0 = 0.01/16
Growth of a spherical seed:
Results are well converged!
ϵ2
= W = M = 1
Δf = 0.3, r*2D
= 0.79
11. Nucleation with noise, convergence with Δt
Δt0 Δt0/4 Δt0/16
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
t 0
t 0
/2
t 0
/4
t 0
/8
t 0
/16
ϵ2
= W = M = 1
Δf = 0.3
Δx0 = 0.4, Δt0 = 0.01
< ξ2
0 > =
0.015
Δx2
0Δt0
12. Nucleation with noise, convergence with Δx
Δx0 Δx0/2 Δx0/4
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
x 0
x 0
/2
x 0
/4ϵ2
= W = M = 1
Δf = 0.3
Δx0 = 0.4, Δt0 = 0.01
< ξ2
0 > =
0.015
Δx2
0Δt0
13. Generating Δx and Δt independent noise patterns
Special technique is used for generating random numbers that ensure similar
noise patterns independent of 𝛥t and 𝛥x:
• For reproducibility, the RNG seed was fixed
• The random numbers were always generated for the finest temporal and
spatial resolutions. The random numbers for the coarser simulations
were obtained from the finer simulations by averaging.
E.g.:
𝜉1 𝜉2
𝜉3 𝜉4
ξ =
1
4
4
∑
i=1
ξi
Var(ξi) = < ξ2
i > ∝
1
(Δx/2)2
Δx/2 Δx/2 Δx
Var(ξ) ∝
2
4
1
(Δx/2)2
=
1
(Δx)2
14. Why no convergence with Δx?
Increasing the spatial resolution when using white noise → new, higher frequencies
are added to the system → the energy of the system is changed (increased) →
nucleation is highly affected
L, N, Δx0 L, 2N, Δx0/2
In fact, for d ≥ 2 the total energy diverges as Δx → 0: ultraviolet divergence
Solution: use a filtered noise with cutoff λc > 2Δx0 before refining the grid
• “Top down” approach: it is just a necessity to have converged solutions, or even
just to avoid the ultraviolet divergence
• “Bottom up” approach: coarse graining with length 𝜆 → fluctuations below 𝜆 are
already included in the system → they should not be added again.
Justification:
15. Convergence, 𝛥x, filtered noise
Δx0 Δx0/2 Δx0/4
Δx0 Δx0/2 Δx0/4
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
x 0
x 0
/2
x 0
/4
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
x 0
x 0
/2
x 0
/4
λc = 2Δx0
λc = 4Δx0
16. approximation (second-order Taylor expansion) of the bare potential around
The result f ð Þ is a renormalised potential for .
These calculations can be readily verified numerically. As an example,
0.95 0.96 0.97 0.98 0.99 1
f
0
0.0002
0.0004
f
Simulation
Theory
Figure 3. Renormalised free energy density of the standard double-well potential as calcula
from Equation (44) and from numerical simulations, for T ¼ 0.05, Dx ¼ 0.5, Dt ¼ 0.005. O
the part close to one of the potential wells is shown. The zero of f was chosen at the minim
of the renormalised potential. The bin size for the histograms was D ¼ 0:01.
40 M. Plapp
ty]at20:5328July2013
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
t 0
t 0
/2
t 0
/4
t 0
/8
t 0
/16
Other issue: the renormalization of F by the noise
Adding noise renormalizes the phase-field equations.
Consider the double well potential around the minima at 𝜙=0,1 → add fluctuations →
asymmetric restoring forces → the mean value vill be shifted from 𝜙=0,1.
Illustration:
my earlier transformation curves
Solution:
Renormalization of the potential
M. Plapp: Philosophical Magazine, 91, 25–44 (2011)
• The properties of our model are not what we think
• No 1 to 1 correspondence between the two nucleation methods (noise EL)
The renormalization is not significant for simulations with large cells (MS instability,
dendritic sidebranching), but cause problems with small cells (nucleation).
17. Method 2: Solving the Euler-Lagrange equations
nonlinear elliptic PDE
δF
δϕ
=
∂f
∂ϕ
− ϵ2
∇2
ϕ = 0
δF
δc
= ∇Mc ∇
∂f
∂c
= 0 →
∂f
∂c
= μ(ϕ, c) = const = μ0
F[ϕ, c] =
∫ [
ϵ2
2
(∇ϕ)2
+ wg(ϕ) + f(ϕ, c)
]
dV
The Euler-Lagrange equations:
Simple binary PF model
with no term(∇c)2
If μ(ϕ, c) is a simple function, then c(ϕ) can be obtained and plugged back into the first ELE
scalar equation
The binary problem is reduced to the single phase-field problem
Solution methods: relaxation methods, shooting methods, etc.
ϕ( ⃗r) → c(ϕ( ⃗r)) → W* = F[ϕ( ⃗r), c(ϕ( ⃗r))]
Further simplification: the spatial dimensions of the problem can be reduced if
spherical or cylindrical symmetry can be assumed
18. Homogeneous nucleation, summary
• Method 1: Add fluctuations (Gaussian random numbers) to the fields and wait…
• Works only on the nanoscale (large undercoolings)
• Add noise directly to non-conserved fields, or as flux noise to conserved fields
• Use the variance determined by the fluctuation-dissipation theorem (as a guide)
• Mind the dimensionality of the problem when considering the scaling with 𝛥x
• Use filtered noise for 𝛥x independent results
• Inhomogeneous systems, complex nuclei, non-trivial nucleation paths are
automatically handled and considered
• Method 2: Insert supercritical seeds at random times at random places
• Best used for small undercooling and larger nuclei
• For quantitative results, determine the solution and its properties via solving the
Euler-Lagrange equations
• Not practical if the system is inhomogeneous
• The consideration of different nuclei is not automatic
Pure homogeneous nucleation is extremely rare in nature!
Because of the renormalization by noise, the results by Method 1 and 2 are not directly
comparable, unless we are at large lengthscales with small noise
19. Heterogeneous nucleation
Classical nucleation theory (CNT):
assumptions:
• sharp interface, bulk properties inside
• isotropic surface energy, spherical
cap
γ
SL
γ
WS
γ
WL ψ
cos( ) =
wl ws
sl
W⇤
2D,het = F2D(r⇤
, ) = S2D( )W⇤
2D,hom
W⇤
3D,het = F3D(r⇤
, ) = S3D( )W⇤
3D,hom
r⇤
= r⇤
het = r⇤
hom
Young equation:
F2D(r, ) = 2r SL + 2r sin ( SW LW) r2
⇡ S2D( ) f
S2D( ) =
( cos sin )
⇡
F3D(r, ) = 2r2
⇡(1 cos ) SL + r2
⇡(1 cos2
)( SW LW)
4
3
r3
⇡ S3D( ) f
S3D( ) =
2 3 cos + cos3
4
S2D(ψ)
S3D(ψ)
0.2 0.4 0.6 0.8 1.0
ψ/π
0.2
0.4
0.6
0.8
1.0
S(ψ)
Results:
S
L
W
20. Phase field modeling of surfaces:
F[ (r), c(r)] =
Z
f( , c) +
✏2
2
(r )2
dV +
Z
Z( ) dS
Free energy functional including the Z(𝜙) surface function:
(Cahn JCP 1977)
At the extremum by 𝜙(r) and c(r), the variation of F should disappear for any
infinitesimally 𝜌(r) and 𝜒(r) compatible with the boundary conditions:
F = F[ (r) + ⇢(r), c(r) + (r)] F[ (r), c(r)] = 0
This leads to the Euler-Lagrange equations
in the volume
on the surface
Cases:
• 𝜙(r) is fixed along the boundary:
𝜌(r)≣0 on the surface, so the
surface EL eq. holds
• 𝜙(r) is not fixed along the
boundary:
the first part of the surface EL
eq. gives the b.c. to use
surface = boundary of the simulation domain → surface properties = boundary conditions
@f( , c)
@
✏2
r2
= 0
@f( , c)
@c
= µ
⇥
Z0
( ) ✏2
r · n
⇤
= 0
J.A. Warren et al. Phase field approach to heterogeneous crystal nucleation in alloys. Physical Review B, 79, 014204 (2009)
21. Model A
(not according to the Hohenberg-Halperin classification!!!)
isosurfaces of
r
surface
Goal: direct realization of the 𝜓 contact angle
(L. Gránásy)
r · n =
r
2w
✏2
(1 ) cos( )
Z0
( ) = ✏2
r · n = 6 SL (1 ) cos( )
Z( ) = SL(3 2
2 3
) cos( )
We need Z(𝜙) to calculate the free energy of the system
22. Model A
László Gránásy
Ni:
•d10-90% = 2 nm
• 𝛾 = 364 mJ/m2
•Δx = 2 Å (1 pixel ~ 1 atom)
•fluctuation-dissipation noise
•thermal feedback
= 45
= 60 = 90 = 120
23. Model A
Solving the PDEs in cylindrical coordinate system (Matlab PDE toolbox)
The work of formation compared to the
classical theory
24. Model B
Constant 𝜙=𝜙0 at the interface (Dirichlet b.c.)
(J. Warren)
Obtaining the 𝜓 contact angle via Young’s law:
wl =
p
2✏2w
Z 0
0
2
(1 )2
= sl(3 2
0 2 3
0)
ws =
p
2✏2w
Z 1
0
2
(1 )2
= sl(1 3 2
0 + 2 3
0)
cos( ) =
wl ws
sl
= 2 2
0(3 2 0) 1
Setting 𝜙=𝜙0 at the interface: wetting layer
0 0.25 0.5 0.75 1
0
45
90
135
180
φ0
ψ
0 0.25 0.5 0.75 1
0
10
20
30
40
φ0
S
crit
There exist a critical value of 𝜙, below which the
interface can freely grow
25. Model B
Solving the PDEs in cylindrical coordinate system
The work of formation compared to the
classical theory
26. Model C
Constant ∇𝜙=h at the interface (Neumann b.c.)
(J. Warren)
wl = Z( 0) + sl(3 2
0 2 3
0)
ws = Z( 0) + sl(1 3 2
0 + 2 3
0)
cos( ) =
(1 4h)3/2
(1 + 4h)3/2
2
Obtaining the 𝜓 contact angle via Young’s law:
−0.15 −0.1 −0.05 0 0.05 0.1 0.15
0
45
90
135
180
h
ψ
r · n =
r
2w
✏2
h
@
@x
= ±
r
2w
✏2
(1 )
1,2,3,4 =
1 ±
p
1 ± 4h
2
27. Model C
Solving the PDEs in cylindrical coordinate system
The work of formation compared to the
classical theory
28. Comparison of Models A, B and C
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
S
W/WSC
Model A
Model B
Model C
All 3 models are in agreement with the classical nucleation theory the in the R → ∞ limit
29. Boundaries with diffuse interfaces
Solution:
• New, diffuse “wall” or “particle” field, 1 inside and 0 outside the excluded region
• New free energy functional and EOMs
Warren et al., Phys Rev. B 79, 014204 (2009)
Possible problems with sharp boundaries:
• Staircase-like approximation of complex geometries
• Sub-pixel movement of the interface is impossible
Justification:
• As the interface width of the particle function goes to zero, the “diffuse” free
energy functional falls back to the “sharp” one → the “diffuse” EOMs should be a
good approximation of the original, “sharp” EOMs
• The same EOMs can be obtained by the mathematically precise Diffuse Domain
Approach (S. Aland, Comput Model Eng Sci 57, 77 (2010))
F =
Z
f( , r ) (1 w) dV +
Z
Z( )|r w| dV
@
@t
= M
F
= . . .
33. Heterogeneous nucleation, summary
• Practically much more relevant, as in real systems, what we almost always observe
is heterogeneous (rather than homogeneous) nucleation
• Our approach to model heterogeneous nucleation is by setting the surface
properties (wetting) via appropriately chosen boundary conditions
• This treatment can be extended to surfaces with diffuse interface (meaning easy
treatment of more complex geometries)
• Nucleation by noise: the same issues as in the homogeneous case
• Nucleation via determining the EL solutions: more difficult than in the homogeneous
case, e.g. no radial symmetry can be utilized
• Lots of possible applications
34. Athermal nucleation
Heterogeneous nucleation: we assumed a large enough surface. But what if the surface
size is limited?
sub-
ryos
aller
i on
dius
ucle-
xist-
avity
cury
oge-
t he
tain-
ated
fore
per-
dent
her-
cata-
areas of nucleant can equally represent the surface
patches considered by Turnbull [15](Fig. 1(a)), or the
active faces of nucleant particles (Fig. 1(b)), for example
the {0001} faces of TiB2 inoculant particles used to
pendent of holding time at a given undercooling, but
proportional to the square of the undercooling DT. An
athermal nucleation law of this type is easy to imple-
ment in numerical modelling of solidification and is
widely used. As proposed by The´voz et al. [10], the
nucleation rate dn/dDT as a function of DT is commonly
taken to have a Gaussian form; this has been used in
probabilistic modelling of realistic grain structures [11].
The form of dn/dDT is not intrinsic to the liquid, but
dependent on thermal history. In particular, as reviewed
by Turnbull [12], the extent to which a melt can be
undercooled may increase strongly with the degree to
which it was superheated above its liquidus temperature
TL. This can be explained as an effect of the survival of
embryos of the crystalline phase above TL in cavities
(conical or cylindrical, in a mould wall or other sub-
strate) [12]. If the superheat is greater, fewer embryos
survive and the survivors are in cavities with a smaller
mouth; the DT at which they become active nuclei on
cooling is inversely proportional to the mouth radius
[12]. (Similar analyses have been applied for the nucle-
ation of gas bubbles in supersaturated liquids. Pre-exist-
ing bubbles in cavities are active nuclei if the cavity
mouth exceeds a critical radius [13].)
Turnbull [14]showed that dispersions of mercury
droplets could be used to measure the rate of homoge-
neous nucleation under isothermal conditions, but he
found that the large DT required was not always obtain-
able. In some dispersions (presumed to be contaminated
with mercury oxide) the droplet undercoolings before
solidification were only 2–4 K [15]. For these disper-
sions, the fraction of droplets solidified was dependent
on DT but not on time. Turnbull attributed this to ather-
mal nucleation at surface patches acting as potent cata-
lysts. He noted that a embryo of the crystalline phase
formed on such a patch could become an active trans-
formation nucleus only when, on cooling, the critical
nucleation radius rÃ
becomes less than the radius of
the patch. On this basis, he was able to derive the size
distribution of patches from the distribution of DT val-
ues at which the droplets solidified.
Similar athermal heterogeneous nucleation occurs in
the solidification of inoculated aluminium alloys [16].
Inoculation with an Al–Ti–B master alloy contributes
particles of TiB2 to the melt, and nucleation on these
is dominant. The particles are hexagonal prisms and
rect predictions of final grain size as a function of refiner
addition level, alloy solute content and cooling rate [16].
The success of this free-growth model has prompted
studies of how the particle size distributions in inocu-
lants might be optimized [18].
In this paper we analyse further the heterogeneous
nucleation of solidification on nucleant substrates of a
defined size. The standard approach to finite-size effects,
taken by Fletcher [19–21], considers nucleant particles of
various shapes, but analyses only the rate of thermal
nucleation under isothermal conditions. We extend this
work by analysing athermal nucleation. For simplicity
we consider nucleant areas that are plane circles of ra-
dius rN, the analysis for more complicated shapes mostly
differing only by geometrical factors. Plane circular
areas of nucleant can equally represent the surface
patches considered by Turnbull [15](Fig. 1(a)), or the
active faces of nucleant particles (Fig. 1(b)), for example
the {0001} faces of TiB2 inoculant particles used to
Fig. 1. Examples (shaded in (a) and (b)) of circular nucleant areas of
the kind considered in this work: (a) a surface patch, (b) the active face
of a nucleant particle. The growth of solid from such a nucleant area
(c) involves an increase in the curvature of the liquid/solid interface
enabled by an increase in undercooling. The curvature is maximum
when the liquid/solid interface is hemispherical and there is free growth
beyond that point. The onset of free growth as the undercooling is
increased constitutes athermal heterogeneous nucleation of
Quested and Greer, Acta Mat. 2005
T = 2 sl
sl : Gibbs-Thomson coe cient
: mean curvatre
small undercooling: large radius
large undercooling: small radius
Athermal process: transformations which are observed not to proceed under isothermal
conditions, but only on cooling
35. Athermal nucleation
g bubbles in cavities are active nuclei if the cavity
outh exceeds a critical radius [13].)
Turnbull [14]showed that dispersions of mercury
oplets could be used to measure the rate of homoge-
ous nucleation under isothermal conditions, but he
und that the large DT required was not always obtain-
le. In some dispersions (presumed to be contaminated
th mercury oxide) the droplet undercoolings before
idification were only 2–4 K [15]. For these disper-
ns, the fraction of droplets solidified was dependent
DT but not on time. Turnbull attributed this to ather-
al nucleation at surface patches acting as potent cata-
ts. He noted that a embryo of the crystalline phase
med on such a patch could become an active trans-
mation nucleus only when, on cooling, the critical
cleation radius rÃ
becomes less than the radius of
e patch. On this basis, he was able to derive the size
tribution of patches from the distribution of DT val-
s at which the droplets solidified.
Similar athermal heterogeneous nucleation occurs in
e solidification of inoculated aluminium alloys [16].
oculation with an Al–Ti–B master alloy contributes
rticles of TiB2 to the melt, and nucleation on these
dominant. The particles are hexagonal prisms and
cleation of solid aluminium is on their flat {0001}
Fig. 1. Examples (shaded in (a) and (b)) of circular nucleant areas of
the kind considered in this work: (a) a surface patch, (b) the active face
of a nucleant particle. The growth of solid from such a nucleant area
(c) involves an increase in the curvature of the liquid/solid interface
enabled by an increase in undercooling. The curvature is maximum
when the liquid/solid interface is hemispherical and there is free growth
beyond that point. The onset of free growth as the undercooling is
increased constitutes athermal heterogeneous nucleation of
solidification.
good wetting properties, easy
heterogeneous nucleation
particle grows as far as the substrate
size allows, when growth must stop
further growth is possible if the radius
can be decreased, i.e., the
undercooling can be increased
critical radius (undercooling)?
Think of pushing a ball through a
hole of radius rN in the wall...
r⇤
= rN
r⇤
! T⇤
T⇤
= T⇤
(rN )
Quested and Greer, Acta Mat. 2005
36. cleant area of radius rN (Fig. 1(c)). The
erfacial energies underlying Eq. (3) does
(in contrast to earlier analyses [19–21]),
fined contact angle. Another consequence
overage of the nucleant area is that line
no roˆle. For the cap, the radius of curva-
quid/solid interface rLS is related to the
ap h by
ð6Þ
f the cap, Vcap is given by
2
À
h3
3
¼ p
r2
Nh
2
þ
h3
6
ð7Þ
f the liquid/solid interface ALS by
¼ p h2
þ r2
N
À Á
: ð8Þ
uired to form a cap of solid (Wcap) has
rom interfacial energies and from the free
associated with the solidification of the
Consistent with there being pre-existing
ence point for energy (Wcap = 0) is taken
n infinitesimally thin layer of solid coating
eant area. The only relevant interfacial en-
ALScLS. Consistent with the derivation of
ee energy of fusion per unit volume is ta-
DT, an excellent approximation for small
of cap formation,
S À pr2
N
Á
À V capDSVDT; ð9Þ
g from Eqs. (7) and (8), can be expressed
The form of Eq. (12) is plotted in Fig. 4 for five values of
dimensionless undercooling. For DT DTfg, the work of
cap formation passes through a minimum followed by a
maximum as h/rN is increased. These extrema occur at
h
rN
¼
DT fg
DT
Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
DT fg
DT
2
À 1
s0
@
1
A ð13Þ
and represent conditions of equilibrium across the li-
quid/solid interface. At these points the radius of curva-
ture of that interface has the critical value rÃ
given by
Eq. (1), expressed in dimensionless terms as
rÃ
rN
¼
DT fg
DT
: ð14Þ
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5 4
Dimensionless cap height ( /
N
)h r
∆ / ∆
fg
= 0.5T T
∆ / ∆
fg
= 0.625T T
∆ / ∆
fg
= 0.75T T
∆ / ∆
fg
= 0.875T T
∆ / ∆
fg
= 1T T
WW*cap
/Dimensionlesscapenergy,∆fg
T
W W*∆
cap
/ ∆ fg
T
Athermal nucleation
Quantitative analysis: calculation of the surface and volume contributions
vature with a plane such that the circle of intersection
has a radius equal to rN. The cap of smaller volume
(a) is in metastable equilibrium; the larger cap (b) is in
unstable equilibrium.
The work of formation and rLS are presented in Fig. 6
for the case of DT/DTfg = 0.5. Between the two equilib-
rium conditions shown in Fig. 5, rLS rÃ
. The work of
cap formation in this figure is normalized with respect
to W Ã
DT the critical work for homogeneous nucleation
at the actual undercooling:
W Ã
DT ¼
16pc3
LS
3DS2
VDT2
: ð15Þ
This normalization would not be useful in Fig. 4, since
W Ã
DT would vary from curve to curve, but it is useful
for the given undercooling in Fig. 6 in indicating how
the work of cap formation compares with the critical
work for homogeneous nucleation.
Fig. 4 shows that for DT/DTfg 1, there is an energy
barrier for nucleation. As the normalization is with re-
spect to the fixed quantity W Ã
DTfg
, the energies on the differ-
ent curves can be compared directly. As the dimensionless
undercooling is increased, the two extrema converge and
the barrier decreases. At DT/DTfg = 1, the work of forma-
tion as a function of cap height no longer exhibits
extrema; there is just the stationary point at
h ¼ rLS ¼ rÃ
¼
2cLS
DSVDT
ð16Þ
when the solid takes the form of a hemisphere and there
is no barrier to free growth (Fig. 4). Cooling through the
condition DT/DTfg = 1 gives athermal nucleation. At
DT/DTfg 1, there could be thermal activation over
the nucleation barrier, and we now assess the likelihood
of this pre-empting athermal nucleation on cooling.
3.2. The competition between thermal and athermal
nucleation
The initial, infinitesimally thin, coating of solid on the
nucleant area grows naturally to the metastable and dor-
mant condition shown in Fig. 5(a). From that condition,
the critical work of thermal nucleation DWcap is the dif-
ference in energy between the two extrema, which can be
expressed in dimensionless terms as
DW cap
¼
DTfg
2
À 1
# ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
DT
2
s
ð17Þ
Fig. 5. The (a) metastable- and (b) unstable-equilibrium configura-
tions of the dormant-solid cap for DT DTfg. The radius of curvature
-0.4
-0.2
0
0.2
0.4
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5 3 3.5 4
rr
Wcap
/Dimensionlesscapenergy,
Dimensionless cap height ( /
N
)h r
∆W W*
cap
/ ∆T
sionlessradiusofcurvature(
LS
/
N
)
Fig. 6. The dimensionless work of formation ðW cap=W Ã
Þ of the solid
cap and radius of curvature of the liquid/solid interface (rLS/rN) as a
function of the dimensionless cap height (h/rN) at a dimensionless
undercooling of 0.5 (Eq. (10)). The work of formation is normalized
with respect to the critical work for homogeneous nucleation at the
actual undercooling (Eq. (15)). A dimensionless undercooling of 0.5
results in a dimensionless critical radius of 2 (dashed line), which is the
value at the extrema on the work curve. The critical work for free
growth (DWcap) is the energy difference between the extrema and is less
than the critical work for homogeneous nucleation.
metastable
unstabe
Quested and Greer, Acta Mat. 2005
Wcap
WΔTfg
= −
1
4
ΔT
ΔTfg (
h
rN )
3
+
3
4 (
h
rN )
2
−
3
4
ΔT
ΔTfg
h
rN
Vcap = π
(
r2
Nh
2
+
h3
6 )
ALS = π(h2
+ r2
N)
Wcap = γLS(ALS − r2
Nπ) − VcapΔSVΔT
For a given h, there are two equilibrium configurations
h
37. Athermal nucleation
and represent conditions of equilibrium across the li-
quid/solid interface. At these points the radius of curva-
ture of that interface has the critical value rÃ
given by
Eq. (1), expressed in dimensionless terms as
rÃ
rN
¼
DT fg
DT
: ð14Þ
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5 4
Dimensionless cap height ( /
N
)h r
∆ / ∆
fg
= 0.5T T
∆ / ∆
fg
= 0.625T T
∆ / ∆
fg
= 0.75T T
∆ / ∆
fg
= 0.875T T
∆ / ∆
fg
= 1T T
WW*cap
/Dimensionlesscapenergy,∆fg
T
W W*∆
cap
/ ∆ fg
T
Fig. 4. Dimensionless work of formation ðW cap=W Ã
DTfg
Þ of the solid cap
as a function of dimensionless cap height (h/rN) plotted for various
values of dimensionless undercooling (DT/DTfg) (Eq. (12)). The work
of formation is normalized with respect to the critical work for
homogeneous nucleation at the free-growth undercooling. The minima
(maxima) in these energy curves represent metastable (unstable)
equilibrium configurations.
ayer covering the nucleant would be energetically less
avourable than the formation of spherical-cap embryos
f solid (Fig. 2(a)).
Conventional analyses emphasise case II and classical
pherical-cap nuclei. Case I has been largely overlooked
ecause it appears to suggest that there is no nucleation
arrier to solidification, even though, as noted in Section
, a wetting layer of solid may be dormant with a free-
rowth barrier to its becoming a transformation nu-
leus. However, case I was considered in early work
n the effect of thermal history on undercooling of liq-
ids; Richards [34]suggested that a crystalline adsorbate
might exist on substrates. This idea was pursued in the
articular case of TiB2 inoculant particles in aluminium,
where the hypernucleation theory considered the condi-
ions for forming a quasi-solid nucleant layer on the sur-
ace of the particles above the liquidus temperature
35,36]. There is indeed some microscopical evidence
or a layer on these particles [17]. Transmission electron
microscopy has recently been used [37]to show that
here can be ordering in liquids at substrate surfaces,
ven above TL.
For the effective nucleation temperature to be given
y Eq. (2) according to the model depicted in Fig. 1(c),
here must be initial formation of solid on the nucleant
ubstrate and either of the cases identified above can ap-
ly. In case I the solid exists as a thin layer even above
TL and there is no nucleation barrier for formation of
his initial solid. It is likely to form even when the cool-
ng rate is very high. In case II, the solid must be nucle-
ted on the substrate and this occurs beyond a critical
ndercooling; accepting that there are geometrical prob-
ems for small h, the spherical-cap model is taken as the
fg
In regime (iii) rN rÃ
and there is no barrier to free
growth; in this case, the rate-limiting step for effective
nucleation (i.e., for the substrate to act as a transforma-
tion nucleus for solidification of the liquid) is the initial
formation of solid on the substrate.
0 0.5 1 1.5 2
ContactAngle,θ
∆ / ∆
fg
T T
0
π/2
(i) (iii)
(ii)
nucleation
not possible
no barrier to
free growth
Fig. 3. Regimes of nucleation behaviour as a function of reduced
undercooling (DT/DTfg) and contact angle h. In (i), the nucleant area is
too small to permit a spherical-cap nucleus to form. In (ii), the regime
of most interest in the present work, a nucleus can form but its growth
stops when it has spread over the nucleant area and thickened such
that the liquid/solid interface reaches the critical radius of curvature r*
.
The case of solid formed by adsorption or wetting can be represented
by h = 0. In (iii), rN r*
, there is no barrier to free growth, and the
rate-limiting step for effective nucleation is the initial formation of
Quested and Greer, Acta Mat. 2005
For ΔT ≃ ΔTfg normal nucleation (fluctuations) can also contribute
38. Athermal nucleation
Dormant embryos. Their size
depend on the undercooling.
T T⇤
(rN )
Dormant embryos start to
grow. No thermal activation is
required!
T = T⇤
(rN )
T T⇤
(rN )
Free growth limit
Use in phase-field models:
Distribute hypothetic particles with a given size
distribution. Induce nucleation (flip a pixel) for
those, that have larger than the critical
undercooling (corresponding to their size)
T = 17K
T = 18K