Nucleation and Growth in Nanosystems: Some New Concepts
Review covers some new concepts in theory and modelling of the initial stages of solid-state reactions in thin films, multilayers, nanoparticles, and bulk nanocrystalline materials. The following topics are included: possibility of oscillatory ordering and nucleation in the sharp concentration gradi...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2004
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| Cite this: | Nucleation and Growth in Nanosystems: Some New Concepts / A.M. Gusak, A.O. Bogatyrev, A.O. Kovalchuk, S.V. Kornienko, Gr.V. Lucenko, Yu.A. Lyashenko, A.S. Shirinyan, T.V. Zaporoghets // Успехи физики металлов. — 2004. — Т. 5, № 4. — С. 433-502. — Бібліогр.: 92 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859715908633624576 |
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| author | Gusak, A.M. Bogatyrev, A.O. Kovalchuk, A.O. Kornienko, S.V. Lucenko, Gr.V. Lyashenko, Yu.A. Shirinyan, A.S. Zaporoghets, T.V. |
| author_facet | Gusak, A.M. Bogatyrev, A.O. Kovalchuk, A.O. Kornienko, S.V. Lucenko, Gr.V. Lyashenko, Yu.A. Shirinyan, A.S. Zaporoghets, T.V. |
| citation_txt | Nucleation and Growth in Nanosystems: Some New Concepts / A.M. Gusak, A.O. Bogatyrev, A.O. Kovalchuk, S.V. Kornienko, Gr.V. Lucenko, Yu.A. Lyashenko, A.S. Shirinyan, T.V. Zaporoghets // Успехи физики металлов. — 2004. — Т. 5, № 4. — С. 433-502. — Бібліогр.: 92 назв. — англ. |
| collection | DSpace DC |
| container_title | Успехи физики металлов |
| description | Review covers some new concepts in theory and modelling of the initial stages of solid-state reactions in thin films, multilayers, nanoparticles, and bulk nanocrystalline materials. The following topics are included: possibility of oscillatory ordering and nucleation in the sharp concentration gradient; criteria of suppression/growth of stable and metastable phases at the nucleation stage; possible nucleation modes in sharp concentration gradient and their competition; competitive nucleation and decomposition in small volumes; criteria of unambiguous choosing the parameters of discontinuous precipitation based on the balance and maximum production of the entropy; formation of nanostructure under uniaxial compression of single-crystalline alloy.
Обзор содержит набор некоторых новых концепций в теории и моделировании начальных стадий твердофазных реакций в тонких пленках, мультислоях, наночастицах и в объеме нанокристаллических материалов. Обсуждаются следующие темы: возможность осцилляционного упорядочения и зародышеобразования в поле градиента концентрации; критерий подавления/роста стабильных и метастабильных фаз на стадии зародышеобразования; возможность различных мод зародышеобразования в поле градиента концентрации и их конкуренция; конкурентное зародышеобразование и распад в малых объемах; критерий однозначного выбора параметров ячеистого распада, основанный на балансе и максимуме производства энтропии; формирование наноструктур под воздействием одноосного сжатия монокристаллического сплава
Огляд містить набір деяких нових концепцій в теорії і моделюванні початкових стадій твердофазних реакцій у тонких плівках, мультишарах, наночастинках та в об’ємі нанокристалічних матеріалів. Обговорено наступні теми: можливість осциляційного впорядкування і зародкоутворення в полі градієнта концентрації; критерій пригнічення/росту стабільних і метастабільних фаз на стадії зародкоутворення; можливість різних мод зародкоутворення в полі градієнта концентрації та їх конкуренція; конкурентне зародкоутворення і розпад в малих об’ємах; критерій однозначного вибору параметрів коміркового розпаду, що базується на балансі та максимумі виробництва
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| fulltext |
433
PACS numbers: 05.70.Np, 64.75.+g, 66.30.Ny, 66.30.Pa, 68.55.Ac, 82.60.Nh, 82.60.Qr
Nucleation and Growth in Nanosystems: Some New Concepts
A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk,
S. V. Kornienko, Gr. V. Lucenko, Yu. A. Lyashenko,
A. S. Shirinyan, and T. V. Zaporoghets
B. Khmel’nyts’ky Cherkasy National University,
Sub-Faculty of Theoretical Physics,
81 Shevchenko Blvd.,
UA-18031 Cherkasy, Ukraine
Review covers some new concepts in theory and modelling of the initial stages
of solid-state reactions in thin films, multilayers, nanoparticles, and bulk
nanocrystalline materials. The following topics are included: possibility of
oscillatory ordering and nucleation in the sharp concentration gradient; crite-
ria of suppression/growth of stable and metastable phases at the nucleation
stage; possible nucleation modes in sharp concentration gradient and their
competition; competitive nucleation and decomposition in small volumes; cri-
teria of unambiguous choosing the parameters of discontinuous precipitation
based on the balance and maximum production of the entropy; formation of
nanostructure under uniaxial compression of single-crystalline alloy.
Огляд містить набір деяких нових концепцій в теорії і моделюванні
початкових стадій твердофазних реакцій у тонких плівках, мультиша-
рах, наночастинках та в об’ємі нанокристалічних матеріалів. Обговоре-
но наступні теми: можливість осциляційного впорядкування і зародко-
утворення в полі градієнта концентрації; критерій пригнічення/росту
стабільних і метастабільних фаз на стадії зародкоутворення; можли-
вість різних мод зародкоутворення в полі градієнта концентрації та їх
конкуренція; конкурентне зародкоутворення і розпад в малих об’ємах;
критерій однозначного вибору параметрів коміркового розпаду, що ба-
зується на балансі та максимумі виробництва ентропії; формування на-
ноструктур під дією одновісного стискання монокристалічного стопу.
Обзор содержит набор некоторых новых концепций в теории и модели-
ровании начальных стадий твердофазных реакций в тонких пленках,
мультислоях, наночастицах и в объеме нанокристаллических материа-
лов. Обсуждаются следующие темы: возможность осцилляционного
упорядочения и зародышеобразования в поле градиента концентрации;
критерий подавления/роста стабильных и метастабильных фаз на ста-
Успехи физ. мет. / Usp. Fiz. Met. 2004, т. 5, сс. 433—502
Оттиски доступны непосредственно от издателя
Фотокопирование разрешено только
в соответствии с лицензией
2004 ИМФ (Институт металлофизики
им. Г. В. Курдюмова НАН Украины)
Напечатано в Украине.
434 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
дии зародышеобразования; возможность различных мод зародышеобра-
зования в поле градиента концентрации и их конкуренция; конкурент-
ное зародышеобразование и распад в малых объемах; критерий одно-
значного выбора параметров ячеистого распада, основанный на балансе
и максимуме производства энтропии; формирование наноструктур под
воздействием одноосного сжатия монокристаллического сплава.
Keywords: concentration gradient, diffusion, nucleation, decomposition,
simulation.
(Received 10 February, 2004)
1. INTRODUCTION (A LITTLE BIT HISTORY)
Present review is an attempt of summarizing the recent endeavours of
modelling and theoretical description of the Big Bang of solid-state re-
actions (SSR) at the very initial stages including nucleation. Evident
nano-trend of materials science makes initial stages of SSR (true nano-
process) an important issue. Our group at the Cherkasy University
treats these problems during last two decades, in co-operation with such
wonderful researchers as C. Gurov and A. Nazarov (Moscow), P.
Desré and F. Hodaj (Grenoble), F. van Loo and A. Kodentsov (Eindho-
ven), K. N. Tu (Los Angeles), G. Schmitz (Muenster), V. V. Slyozov and
L. N. Paritskaya (Kharkiv). Some ideas and new understanding in this
field, which nucleated, grew and even partially ripened during this pe-
riod, include (1) ordering, nucleation, competition and growth of sta-
ble and/or metastable intermediate phases in the nanoscale diffusion
zone–phase transformations in sharp, time-dependent concentration
gradient, (2) competitive nucleation in the isolated binary nanoparti-
cles or bulk samples with very high density of nucleation sites, (3)
flux-driven coarsening in nanosystems, (4) possibility of bulk nano-
crystalline alloy formation under pulse loading.
So, how does the ‘reactive Big Bang’ look like?
Till 80’s a diffusion community, occupied mainly with diffusion and
reactions in macrospecimens, treated the initial stages of these proc-
esses as some exotic problem, basically, as theoretical one. Reason was
that initial stage is usually being ‘forgotten’ by the system (in accor-
dance with basic principles of non-equilibrium thermodynamics and
statistical mechanics) when the diffusion zone reaches several mi-
crometers.
In case of interdiffusion with full mutual solubility, the notion ‘ini-
tial stage’ means the period till reaching the parabolic regime both for
concentration redistribution and for Kirkendall shift. It includes:
1. Formation of non-equilibrium vacancy distribution due to va-
cancy flux divergence arising from different mobilities of spe-
cies, and its relaxation due to vacancy sources/sinks [1—4].
Nucleation and Growth in Nanosystems: Some New Concepts 435
2. Possible diffusion-induced recrystallization–formation of new
grains of solid solution [5].
3. In case of polycrystalline materials, transition from pure grain
boundary diffusion (regime C) to the Fisher regime (B), and
then to parabolic regime (A) [6].
In case of reactive diffusion, ‘initial stage’ means the period till
formation and parabolic growth of all stable intermediate phases. It
includes, in principle, the following sub-stages:
1. Nucleation of intermediate phases–formation of the new phase
overcritical nuclei (islands) at or in the vicinity of initial contact inter-
face [7].
2. Growth of new phase islands and competition (for space and mate-
rial) between them till formation of the phase layers with more or less
planar geometry. In case of ternary and multicomponent systems, the
two-phase layers can as well be formed.
3. Overcoming of interface barriers (if they exist) [1, 8—10] and/or
the relaxation of vacancy subsystem.
Practical interest in the initial stages of SSR appeared due to inves-
tigation of reactive diffusion in thin films. The usually observed one-
by-one (sequential) phase growth meant that the initial stage for some
phases might well be simultaneously the final stage. If, for example,
intermediate phase A1B2 appeared to be suppressed by some reasons
during the growth of the phase A2B1 till the consumption of pure B,
this phase will never appear at all. Yet, even for explaining phase sup-
pression, people tried to avoid the consideration of nucleation proc-
esses. Instead, the suppression has been explained in terms of interplay
of diffusion and interfacial barriers. Most successful theory of this
type belongs to U. Goesele and K. N. Tu [11]. Introducing some arbi-
trary rate constants for fluxes across interfaces, they managed to de-
rive expressions for the critical thicknesses of suppressing phases and
delay periods of suppressed phases.
Nucleation issues, in terms of standard nucleation approach, but in
application to solid-state reactions, have been discussed by F. d’Heurle
in 1986 [7].
Situation became even more intriguing with discovery of the solid
state amorphizing reactions (SSAR) [12, 13] demonstrating the growth
of metastable amorphous layer without any evidence of stable phase
formation till the amorphous layer reached certain critical thickness of
hundreds of nanometres. It looked like the nucleation and/or growth of
stable phases have been suppressed not by just low temperature, but by
the growing metastable phase.
Detailed DSC-investigations of solid state reactions in multilayers
by the groups of K. Barmak and P. Gas demonstrated [14—16] the pos-
sibility of the two heat release maxima for the same phase, which
means some kind of two-stage phase formation (the first stage, possi-
436 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
bly, being the lateral growth of new phase islands).
Recently G. Schmitz et al. used an atom-probe tomography (TAP)
method to investigate the very initial stage of solid-state reactions [5,
17]. This method, with resolution of a few angstroems, provides us
with the 3-dimensional distribution of atoms of the reacting species.
These experiments clearly show that usually even the first islands of
the first phase to grow do not form immediately after contact.
One more important new nano-field governed by nucleation is the
formation and properties of bulk nanocrystalline materials. Simulta-
neous and fast nucleation in highly imperfect supersaturated solid so-
lutions yields very small supply regions capable to feed new phase em-
bryo at each nucleation and growth site. As an example, the so-called
bulk metallic glasses (BMG) [18, 19] often devitrify with a very high
nucleation rate yielding, in a first crystallization stage, to a dispersion
of nanocrystals in the amorphous matrix [20]. Arguing that such a
high frequency of nucleation cannot be due to heterogeneous nuclea-
tion, Kelton [21] has explained such a phenomenon by enrichment with
the crystallizing component in a shell, (of atomic size width), sur-
rounding embryos. During quenching, those frozen subcritical em-
bryos, with their surrounding enriched shell serving as a feeding ma-
terial, become overcritical and practically stop growing after consum-
ing the shell component.
Thus, indeed, the ‘nano-vector’ of solid-state reactions makes it
more and more important to take the nucleation stage into account.
In 1982, one of the authors (A.G.), jointly with his teacher, late
Prof. Cyril Gurov (A. A. Bajkov Institute of Metallurgy, Moscow) pre-
sented a simple (even ‘naïve’, as we see it today) model of the phase
competition taking into account the nucleation stage of each phase
[22]. Actually, the only concept, which had been taken in [22] from the
nucleation theory, was the existence of critical nuclei. They appear due
to some miracle called the heterophase fluctuations, which are the sto-
chastic events and cannot be described by some deterministic model.
Initial idea was just that each phase cannot start from zero thickness;
it should start from some critical size particle (about nanometre). Con-
trary to standard nucleation theory, the critical nuclei of intermediate
phases during reactive diffusion are formed in the strongly inhomoge-
neous region–interface between other phases. Therefore, from the
very beginning they have to let the diffusion fluxes passing through
themselves. Evidently, fluxes change abruptly when passing across
each new-formed boundary of the newly born nucleus, and thus drive
the boundary movement. This picture of interface movement due to
flux steps is well known for diffusion couples under the name of
Stephan problem and means diffusive interactions between neighbour-
ing phases. Yet, the initial width of each phase is taken to be the criti-
cal nucleus size (instead of zero). Peculiarity of initial stage is just the
Nucleation and Growth in Nanosystems: Some New Concepts 437
possibility that the width of some phase nucleus (distance between left
and right boundaries can decrease as well as increase. If it decreases,
then the nucleus becomes subcritical and should disappear. Usually it
happens if the neighbouring phases have larger diffusivity and com-
parative thickness. Then these neighbours (‘vampires’ or ‘sharks’) will
destroy and consume all of the new-forming nuclei, making new phase
to be present only virtually–in the form of constantly forming (due to
heterophase fluctuations) and vanishing (due to diffusive suppression
by the neighbours) embryo.
Simple mathematical scheme was built predicting the sequence of
phase formation and incubation periods, provided that one knows the
integrated Wagner diffusivities and critical nucleus size for each
phase. Simplest example of this scheme is presented in Section 3. This
scheme was applied to competition with solid solutions [23], to the
phase growth under strong electric current [24], to the reactive diffu-
sion in ternary systems [25], and to the phase competition in reacting
powder systems [26, 27]. Applications to electric field case demon-
strated that the phase spectrum of reaction zone can be influenced and
even controlled by strong enough current density. Large current densi-
ties become real due to miniaturization of integrated schemes and in-
troducing of the flip-chip technology [27, 28]. Last results on reactive
diffusion in UBM-solder contacts under strong current crowding [29]
confirm the mentioned idea: without current or under weak current,
the reaction between copper and dilute solution of tin in lead at 150°C
demonstrates only Cu3Sn1-phase formation; current density of
8 210 A/mj > leads to formation and fast growth of Cu6Sn5 phase.
Later we realized that in our naпve model of the phase competition
we had taken the inhomogeneity of nucleation region into account only
partially. Namely, we treated the diffusive interactions of the newborn
nuclei, but we had not considered the possible change of the nucleation
barrier, size and shape, caused by the very fact of sharp concentration
gradient. Thus, we had to reconsider the thermodynamics of nuclea-
tion in the concentration gradient. The very first version of such a the-
ory was presented at the conference ‘Defects and Diffusion’ (DD-89) in
Russia and first published May 1990 [30] (see also [31]). Main idea was
as follows: if, prior to intermediate phase formation, narrow layer of
metastable solid solution or amorphous alloy had been formed at the
base of initial interface, the sharp concentration gradient inside this
layer provides decrease of the total bulk driving force of nucleation,
and corresponding increase of nucleation barrier. Nuclei were taken to
be spheres, appearing in the strongly inhomogeneous concentration
profile of the parent phase, so that local driving force of transforma-
tion could change significantly from the left to the right along the di-
ameter of nucleus. This effect appeared to be non-negligible, since
usually the intermediate phases have very strong concentration de-
438 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
pendence of the Gibbs energy. Main result was the new size dependence
of the Gibbs energy; it contained, in addition to the terms of second or-
der (surface energy, positive) and third order (bulk driving force,
negative), the new term proportional to the 5th power of size and the
squared concentration gradient:
( )22 3 5( )G R R R C R∆ = α − β + γ ∇ , (1.1)
γ being positive and proportional to the second derivative of the new
phase Gibbs energy over concentration. Expression (1) means that, for
large enough gradient
critC C∇ > ∇ (typically
crit 8 110 mC −∇ ∝ ), the de-
pendence ( )G R∆ becomes monotonically increasing (infinitely high
nucleation barrier) meaning thermodynamic suppression of nucleation
by the very sharp concentration gradients. Thus, according to our
model, at the very initial stage of reactive diffusion the nucleation can
be suppressed even without diffusive competition, just due to too nar-
row space region, favourable for transformation.
Independently, similar results were published by P. Desré et al. in
1990, 1991 [32, 33]. This approach had been applied to description of
solid-state amorphizing reactions [32—35], explaining why the stable
intermetallics appear in diffusion zone only after amorphous layer ex-
ceeds some critical thickness.
In spite of similar results, models [30, 31] and [32, 33] of nucleation
in the sharp concentration gradient treated quite different possible
mechanisms (nucleation modes). In [30, 31], a polymorphous mode has
been suggested according to the following picture. Initial diffusion
leads to formation and growth of metastable parent solution with
sharp concentration profile. When this profile becomes smooth enough
to provide sufficient space for compositions favourable for new inter-
mediate phase, this very phase nucleates just by reconstruction of
atomic order, without changing immediately the concentration profile
(at ‘frozen’ diffusion)–polymorphic transformation. In [32, 33] the
transversal nucleation mode was suggested bearing in mind the follow-
ing picture: each thin slice of the new-formed nucleus, perpendicular
to direction of concentration gradient, is considered as a result of de-
composition in corresponding thin infinite slice of parent solution,
leading, of course, to redistribution of atoms among new and old
phases. In this transversal mode, the redistribution proceeds within
each slice, independently on others.
In Ref. [36], one more mechanism has been suggested (and analyzed
in more details in Refs. [37, 38])–total mixing (longitudinal) nuclea-
tion mode, when the redistribution of atoms proceeds during nuclea-
tion, but only inside the new forming nucleus. Contrary to the two pre-
vious modes in this case, the concentration gradient assists the nuclea-
tion–in expression (1), coefficient γ is negative.
Nucleation and Growth in Nanosystems: Some New Concepts 439
Above-mentioned approach was generalized taking into account the
shape optimization [39—42], the stresses [43], ternary systems [44],
heterogeneous nucleation at grain boundaries [41], at interphase inter-
faces [45]. Most simple models in the frame of this approach are pre-
sented in Section 4.
‘Natural’ thing is to expect that nature will use the mechanism with
lowest nucleation barrier–the total mixing mode. Yet, nucleation is
ruled not only by thermodynamics but by kinetics as well. Thermo-
dynamics of nucleation with constraints indicates only some probable
paths of evolution. Real path is chosen by kinetics, taking into account
not only the free energy profit, but as well the different ‘mobilities’
along each path. Mobilities often appear to be inverse to the profit that
is kind of compensation rule, analogous to relation between activation
enthalpy and frequency factor in diffusion.
To calculate the nucleation kinetics, we used the Fokker—Planck ap-
proach, first applied to nucleation problems by Farkas [46] and recog-
nized after classical work of Zeldovich [47]. Our contribution to this
approach was just taking into account that the driving force depends
on concentration gradient, which in its turn depends on time according
to diffusion laws. It has been shown in [38] that the relative contribu-
tion of each mechanism depends on the ratio of atomic mobilities in the
parent and nucleating phases. If atomic mobility in the new phase is
much lower than in parent one, we can forget about total mixing mode.
If opposite (high mobility inside new phase), then nucleation will pro-
ceed via total mixing, very fast (‘fast is the first’). Main results are
briefly reviewed in Section 5. One of the ‘raisins’ of total mixing (as-
sisting) mode was that the easily formed nuclei, if not growing too
fast, in comparison with decrease of concentration gradient, after
some period can find themselves to be subcritical and be destroyed.
Further natural question was ‘How does system proceed from nu-
cleation of isolated particles to formation of continuous layer, which
even initially appears much thicker (typically about 8 nm [48]) than
the typical nucleus size?’ Model of ‘almost lateral’ growth, driven by
interface diffusion along the curved moving new phase boundaries, is
briefly discussed in Section 6.
After discovering that concentration gradient can play a role of con-
straint on nucleation, it was natural for us to look for other constraints.
Sharp concentration gradient means narrow layer, suitable for nuclea-
tion, e.g. limited volume. Therefore, the most natural thing was to treat
the nucleation in small (nanosize) particles and the multiple simultane-
ous nucleations during formation of bulk nanocrystalline materials.
Main result was as follows: depending on the volume of parent particle
(or, in bulk, on the volume of ‘responsibility region’ around the nuclea-
tion site), the same 3 possibilities exist as in sharp concentration gradi-
ent–nucleation and growth (large size), metastable state (medium
440 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
size), forbidden nucleation (small size). Results were published in [49,
50]. Afterwards we learned that similar results have been published by
Rusanov 30 years before [51] and later developed in [52]. Yet, we hope
that our application of these ideas to the ‘traffic jam’ effect in bulk
glasses might be of some interest (Section 7): When many persons try to
pass through narrow door simultaneously, the process stops. Similar
effect can be responsible for long-living nanocrystalline states [53].
Most of above-mentioned problems are related to choosing a path-
way for the evolution of a non-equilibrium system exhibiting phase
transformations, whereby various regimes are possible for the same
initial and boundary conditions. One more problem of this type, involv-
ing an invariant in the form of a product of the squared period of the
growing structure and the growth rate, is how to predict both of these
parameters for a discontinuous precipitation of a binary alloy with a
given composition, which is supersaturated as a result of supercooling
[6, 54]. In contrast to a spinodal decomposition that takes place homo-
geneously in the whole volume and is controlled by changes in the elas-
tic energy [55, 56], the transformation region represents a moving
large-angle incoherent grain and phase boundary. A model of discon-
tinuous precipitation in supercooled binary polycrystalline alloys at
reduced temperatures, taking place because of the diffusion-induced
grain-boundary migration, is constructed with allowance of grain
boundary diffusion (Section 8). The approach based on the balance and
maximum production of the entropy allows independent determination
of the main parameters, including the interplate distance, the maxi-
mum velocity of the phase transformation front, and the concentration
jump at this boundary [57].
One more unexpected field of ‘nano-ideas’ is a so-called anomalous
mass-transfer under pulse loading of metals–transfer of atoms at dis-
tances of several microns, and sometimes much more, during shock
lasting for less than hundred microseconds [58, 59]. Our recent mo-
lecular-dynamics simulation demonstrates the possibility of nanograin
structure formation during the shock wave propagation in single crys-
tals (Section 9). This result correlates with recent experimental results
on shock loading of monocrystalline copper [60]. Existence (even vir-
tual, during the shock) of nanograin structure can assist the mass-
transfer via mechanisms of grain-boundary or (that is more probable)
‘mechanical diffusion’ [61]. In Section 10, we discuss briefly some pos-
sible future developments.
2. KINETIC MC SIMULATIONS
2.1. Direct MC Simulation of Reactive Diffusion
Since in-situ observations of intermediate phase nucleation are lacking
Nucleation and Growth in Nanosystems: Some New Concepts 441
so far (to our knowledge), let us see what simulations can do.
Recently, we investigated the kinetics of ordering during interdif-
fusion and found the possibility of oscillatory ordering [62] (also see
subsection 2.2). In that case, ordering was a second-order transition.
Simulation of first order transitions in square lattice has some diffi-
culties. If one takes into account only the nearest neighbours interac-
tion, then transition to B2-ordered phase is of second-order type: or-
dering without nucleation barrier. The formation of ordered interme-
diate phase is often related to change of electron subsystem, meaning
the change of effective pair potentials of atomic interactions. To simu-
late first-order transitions, including nucleation of intermediate phase
we modified our model making pair interactions extremely dependent
on the local order. The simplest (and so far the most effective) model is
as follows.
Model system is a 2-dimensional square lattice filled by A and B at-
oms with initial step-like distribution ( BLC , BRC ) and periodic bound-
ary conditions transforming diffusion couple into multilayer. Atoms A
are regarded as ‘special’ (and denoted as A
*) if they are surrounded by 4
neighbouring atoms B. The same is for B
*. These ‘special’ atoms have
‘special’ interactions with their neighbours. Our choice was as follows:
3.0AA BB
kT kT
Φ Φ= = − , 3.225AB
kT
Φ = − ,
* *
0.5AB A B AB
kT kT kT
Φ Φ Φ= = + ,
* *
2.0A B AB
kT kT
Φ Φ= − .
Migration of atoms was induced by vacancy migration. For each va-
cancy position, activation energies of exchange with each nearest
neighbouring (NN) and next nearest neighbouring (NNN) atom were
calculated as differences iE∆ between the saddle-point energy and the
energy in equilibrium site before jump. To simplify the model, we took
all saddle-point energies to be equal to zero. Probability of each of 8
possible NN and NNN vacancy jumps was calculated as
8
1
i
j
E
kT
E
kT
j
e
p
e
∆−
∆
−
=
=
∑
. (2.1)
Subsequent MC simulation, using the residence-time algorithm [63],
gave us full information about position of each atom after each MC
step (MCS), including concentration profiles (averaged over y for each
x-plane), domains of full-order A
*B*-particles of new phase, time de-
pendence of ordered volume and of the system’s energy [64].
Here, we show some results for two initial concentration steps–
442 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
(0.05—0.95) and (0.30—0.70) (Fig. 2.1).
1. For big initial concentration step, a new phase growth has clearly
oscillatory behaviour–antiphase domains of intermediate ordered
phase appear and disappear, competing with each other and with par-
ent phases. At the first stage, actually, all appearing nuclei eventually
disappeared. Yet, with time some of them started surviving, which
corresponds to our old naпve model of phase competition (the vampire
phase being just the inhomogeneous solid solution with diffusivity
much more than in the intermediate phase). In our model each domain
is not permeable for vacancies, so that growth of new phase slows down
after forming an initial layer, and further growth is possible due to
‘channels’ between antiphase domains.
2. Growth of new phase is faster for less initial concentration step.
This result coincided with our theoretical prediction about suppression
of new phase by the sharp concentration gradient [30, 31].
3. Phase growth clearly demonstrates the lateral regime, predicted
by Coffey, Barmak et al. [14].
2.2. Direct MC Simulation of Interdiffusion with Ordering
It is interesting also to understand, how the interdiffusion proceeds at
Fig. 2.1. Ordered intermediate phase formation in multilayer with initial
concentration steps CR − CL = 0.90 and 0.40.
Nucleation and Growth in Nanosystems: Some New Concepts 443
the very initial stage, if ordered phase does not need to overcome any
nucleation barrier. In that case, we can observe the interplay of inter-
diffusion and ordering in concentration gradient. So, we used the same
residence-time algorithm with constant pair interactions between the
nearest neighbours in two-dimensional square lattice (triangular lattice
was investigated as well). Energy of atom before jump was taken just as
the sum of three energies (fourth neighbour being the vacancy). In the
saddle-point of the jump to the nearest site, the atom is interacting with
four neighbours. Our choice for pair energies was as follows:
for atoms in their sites, 3AA BB
kT kT
Φ Φ= = − , 5AB
kT
Φ = − ;
for jumping atom in the saddle point, , 0.5ij ijB B′Φ = ⋅ Φ = .
Initial distribution of atoms corresponded to diffusion couple with
different concentrations on the ‘left’ and ‘right’–concentration step.
Initial profiles with concentrations steps 0.00—1.00, 0.20—0.80, 0.40—
0.60 and 0.50—0.50 (homogeneous case) were simulated.
Results were a little bit unexpected (for details see [62]):
1. Formation of ordered domains in the couples with nonzero con-
centration steps clearly demonstrated nonmonotonic behaviour–
blinking. Domains appeared, lived for some time and then disappeared.
Blinking was more pronounced for samples with larger initial concen-
tration steps.
2. The total area of ordered phase, of course, eventually grew with
time, but not monotonically: the time dependence of the ordered re-
gion’s area with order parameter above 0.9 showed nonmonotonic sto-
chastic behaviour. To distinguish the usual statistical noise of MC pro-
cedure from real order oscillations, we used the coarsening of the time
scale. Choice of coarsening scale was made in such a way, that it pro-
vided monotonic ordering of initially homogeneous samples (couples
0.5—0.5). The results are shown in Fig. 2.2.
To explain these rather unexpected results, we proposed the follow-
ing picture. Diffusivity strongly depends on the order parameter.
Therefore, the newly formed ordered regions, though energetically fa-
vourable, have substantially less diffusivity than the surrounding par-
ent disordered solid solution. Hence, they start to ‘loose their game’ in
diffusive competition with neighbouring phase. Therefore, we can ob-
serve the competition of two factors–thermodynamic profit and ki-
netic disability leading to oscillations. We tried to check, if such ex-
planation is reasonable, by means of the following simplified phe-
nomenological model of simultaneous interdiffusion and ordering with
nonlinear dependence of diffusivity on the order parameter:
( )
c c
D
t x t
∂ ∂ ∂= η ∂ ∂ ∂
% , (2.2)
444 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
( )( )e c
t
∂η = −γ η − η
∂
, (2.3)
where ( )e cη is an equilibrium value of order parameter η correspond-
ing to local value of concentration.
Dependence ( )D η%
was proposed as quadratic under exponent, in the
frame of mean-field considerations for the energy of jumping atoms in
the sites:
( )2
0( ) expD Dη = −αη% (2.4)
with α being proportional to the mixing energy.
Fig. 2.2. Time dependence of the ordered ‘volume’ fraction (with absolute
value of order parameter larger than 0.9) in the planar layer of the diffusion
zone containing 40 atom rows. Time is measured in Monte Carlo steps (MCS).
Nucleation and Growth in Nanosystems: Some New Concepts 445
Numeric solution of the problem (2.2)—(2.4) demonstrated a possi-
bility of nonmonotonic ordering for the certain region of parameters
D0, γ, α. Characteristic time dependences of ordered (with 0.3η > ) re-
gion’s size for dimensionless values D0 = 1, γ = 1, α = 3, 5 are shown in
Fig. 2.3. Increase of parameter α leads to more pronounced oscillation
of order. These oscillations depend also on the initial concentration
step: the larger is this step, the more pronounced are the oscillations
(similar to MC simulations); see Fig. 2.4.
Thus, the main results of phenomenological model coincide with MC
results, except the number of oscillations. We believe the reason is just
one-dimensional character of the phenomenological formulation.
Fig. 2.3. Time dependences of ordered region’s size (η > 0.3) for D0 = 1;
γ = 1; α = 3, 5. Initial concentration step is 0.00—1.00.
Fig. 2.4. Time dependences of the ordered region’s size for different initial
concentration steps; D0 = 1, γ = 1, α = 5.
446 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
3. PHASE COMPETITION
As we have just seen in MC simulations, a phase growth in diffusion
zone starts with the nucleation of intermediate phases as a result of
heterophase fluctuations. A nucleus is thermodynamically stable if its
size exceeds the critical value (if the gain in bulk free energy due to at-
tachment of monomers begins to exceed the loss in surface energy).
The fundamental difference from the phase formation by alloy cooling
is as follows: the phase nuclei are formed in the field of chemical-
potential gradients, and therefore, at once begin to interact diffusion-
ally with parent phases and/or with the nuclei of other intermediate
phases. This diffusional interaction is mathematically described by the
system of balance equations for the fluxes on the moving interphase
boundaries. As a result, the growth velocity of some critical nuclei ap-
pears to be negative, they start decreasing, transforming into embryos
with subcritical size. Such embryos are thermodynamically unstable
and must dissociate. It means that certain intermediate phases from
the phase diagram of the system can be absent in the diffusion zone for
some time (which can be a rather long time). To be more precise such
suppressed phases are virtually present, in the form of forming, de-
creasing and dissociating nuclei. The corresponding criteria of phase
growth/suppression were obtained.
3.1. Simple Phenomenological Model of Diffusive Phase Competi-
tion of the Two Phases
Consider binary diffusion couple A—B with two intermediate phases 1,
2, and negligible solubilities of A in B and of B in A. If, by some rea-
sons, only phase layer 1 grows in diffusion zone, then its growth law is:
( )
1 1
1
1 1
2
1
D C
x t
C C
∆∆ =
−
(3.1)
(∆C1–concentration range of the 1-st phase, effective diffusivity
( )
1
1
1
1
C
D D C dC
C ∆
≡
∆ ∫ %
[61, 62]), ( )D C% –interdiffusion coefficient).
Since the reaction 1 + B → 2 is thermodynamically favourable, the
phase 2 should form the critical nuclei with longitudinal size 2l at the
moving interface 1 − B due to heterophase fluctuations. As will be
shown below, sharp concentration gradients lead to the plate-like
shape of the nuclei. Therefore, one can consider left and right bounda-
ries of the nucleus as nearly flat. Steps of the diffusion flux profiles at
these boundaries generate their movement. According to conservation
Nucleation and Growth in Nanosystems: Some New Concepts 447
of matter at the moving boundaries,
( ) 2 1 1 2 2
2 1
1 2
Ldx D C D C
C C
dt x l
∆ ∆− = −
∆
, ( ) 2 2 2
2
2
1 0Rdx D C
C
dt l
∆− = − (3.2)
(we neglect solubility of A in B and assume the regions of homogeneity
to be narrow).
Hence, growth/shrinkage rate of critical nucleus’ width is equal to
2
22 2 1 1 1 2 2
2 1 1 2 2
11
1
R L
l
dxd x dx D C C D C
dt dt dt C C x C l
∆ ∆ − ∆= − = − + − ∆ −
. (3.3)
One can easily see that this expression is positive for big ∆x1, but can be
negative for small 1x∆ .
If
* 2 1 1
1 1 2
1 2 2
1
1
C D C
x x l
C D C
− ∆∆ < ∆ =
− ∆
, then
2
2 0
l
d x
dt
∆ < , so that every critical
nucleus decreases (being ‘eaten’ by rapidly growing neighbouring
phase 1) and therefore becomes subcritical embryo (unstable), and
should be dissolved.
Such unsuccessful attempts of phase-2 nucleation will be repeated
during some ‘incubation period’ τ, till the suppressing phase 1 reaches
thickness
*
1x∆ growing according to parabolic law (3.1):
( ) ( ) ( )
( ) ( )
2
21 1 1 2* 21 1
1 22
1 1 1 2 2
1 1
2 2 1
C C C C D C
x l
D C C D C
− − ∆τ = ∆ =
∆ − ∆
. (3.4)
Of course, Eq. (3.4) can be used only if the diffusion couple is suffi-
ciently large. If the specie B is consumed before phase-1 layer could
reach the critical thickness
*
1x∆ , the phase 2 will never appear at all.
So far, we just assumed that it is phase 1, which grows first. To de-
termine which of phases will actually grow first, one should consider
diffusive interaction between two initial layers of critical nuclei of
both phases. One can easily check that
1
1 2 1 1 2 2
2 1 1 1 2
1
l
d x C D C D C
dt C C C l l
∆ ∆ ∆= − −
,
2
2 1 1 1 2 2
2 1 1 2 2
11
1l
d x D C C D C
dt C C l C l
∆ ∆ − ∆= − + − −
. (3.5)
System has three possibilities:
1)
1
1 1 2 1 1
2 2 1 2
0
l
D C l C d x
D C l C dt
∆ ∆< ⇒ <
∆
,
2
2 0
l
d x
dt
∆ > , i.e. phase 2 (‘vampire’)
448 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
starts growing, ‘eating’ the nuclei of the phase 1;
2)
1 2
1 1 1 2 1 1 2
2 2 2 1 2
1
0, 0
1 l l
C D C l C d x d x
C D C l C dt dt
∆ − ∆ ∆
< < ⇒ > >
∆ −
, i.e. both phases
grow from the moment of nucleation;
3)
1 1 2 1
2 2 1 2
1
1
D C l C
D C l C
∆ −>
∆ −
, i.e. phase 1 (‘vampire’) starts growing, ‘eating’
the nuclei of phase 2.
Phase suppression cannot continue infinitely. As the suppressing
(‘vampire’) phases grow, the concentration gradients and correspond-
ing fluxes along the phase layers decrease, their growth velocity slows
down and their ‘competition ability’ also decreases, For every sup-
pressed phase the moment exists (see Eq. (3.4)), when the growth ve-
locity of its critical nuclei becomes positive, and they start to grow.
The time of diffusion suppression of the phase nuclei is called the incu-
bation period. Evidently, it is only the ‘diffusion’ part of the full incu-
bation period. The process mentioned above was called diffusion phase
competition. Such a simple approach is easily generalized on the arbi-
trary number of intermediate phases [31].
4. THERMODYNAMICS OF NUCLEATION IN CONCENTRATION
GRADIENT
So far, we used classic nucleation theory. Now, we reconsider thermo-
dynamic of nucleation in the contact zone.
4.1. Polymorphous Nucleation Mode
Let the nucleus of intermediate phase appear in the frozen concentra-
tion profile formed by interdiffusion in the metastable continuous
phase (solid solution or amorphous phase) (Fig. 4.1). If concentration
gradient is sharp enough (narrow diffusion zone), the driving force of
nucleation per atom ag∆ (function of composition) becomes the func-
tion of space coordinates as well.
Hence, the change of Gibbs free energy caused by formation of nu-
cleus in such sharply inhomogeneous conditions is a sum of different
contributions from each thin slice ( )S x dx and is given by the following
expression:
( )( ) ( )( )( ) ( )new oldG n g C x g C x S x dx S∆ = − + σ∫ . (4.1)
Here, we neglect the volume changes (atomic density 1 2n n n≈ = ) and
the corresponding stresses. S is the area of newly born interface sur-
face, σ–its surface energy per unit area, S(x)–area of nucleus cross-
Nucleation and Growth in Nanosystems: Some New Concepts 449
section with plane x, perpendicular to concentration gradient, g–
Gibbs free energy per one atom. To make mathematics simpler, the fol-
lowing approximations are suggested:
( ) ( ) ( ) ( )2 2
0 0 0 0,
2 2
old new
old old old new new newg C g C C g C g C C
α α= + − = + − (4.2)
( ) ( )0C x C x C≅ + ⋅ ∇ . (4.3)
Taking into account Eqs. (4.2), (4.3), one obtains:
( )( ) ( )2 2
0 1 2G n A A C x A C x S x dx S∆ = + ∇ ⋅ + ∇ + σ∫ ,
where
Fig. 4.1. ‘Gibbs free energy versus composition’ and corresponding ‘composi-
tion versus diffusion co-ordinate’ dependences. Driving forces per atom of
nucleus are indicated for (a) polymorphous, (b) transversal and (c) total mix-
ing modes.
450 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
( )( ) ( )( )
( )( ) ( )( ) ( )
2 2
0 0 0 0 0
1 0 0 2
1
0 0 ,
2
0 0 , 2.
old new old old new new
new new old old new old
A g g C C C C
A C C C C A
= − − + α − − α −
= α − − α − = α − α
(4.4)
4.1.1. Spherical Nuclei
Let nucleus be the sphere with the centre in some point Cx , so that
( ) ( )( )22
CS x R x x= π − − .
Then simple algebra transforms Eq. (4.1) into
2 3 5
0 2
4 4
4
3 15
G R n B R B R
∆ = σ ⋅ π + π +
, (4.5)
where
( ) ( )2 22
0 0 1 2 2 2, C CB A A x C A x C B A C= + ∇ + ∇ = ∇ . (4.6)
First of all one should find an optimal place for nucleation from the
conditions
0
C
G
x
∂∆ =
∂
,
2
2 0
C
G
x
∂ ∆ >
∂
:
( )( ) ( )( )
( )
0 01
2
0 0
2
old old new new
C new old
C C C CA
x
A C C
α − − α −
= − =
∇ α − α ∇
(4.7)
(it corresponds to the minimum, if
new oldα > α ).
Thus, with changing concentration gradient time the optimal place
of nucleation shifts but the corresponding concentration in the centre
( ) ( ) 1
2
0 0
2C
A
C x C C
A
+ ∇ = −
remains the same.
Further, we restrict ourselves only to nuclei forming in the optimal
place. In this case, the size dependence of G∆ has a simple form:
( ) ( )22 3 5G R R R C R∆ = α − β + γ ∇ , (4.8)
Nucleation and Growth in Nanosystems: Some New Concepts 451
where
( ) ( )
2
2
1
0 0 0 0 0
2
4
4 , ,
15 2
4 4
.
3 4 3 2
new old
new old
old new old new
new old
n
An n
A g g C C
A
π α − αα = πσ γ =
π π α − α β = − = − + − α − α
(4.9)
Coefficient 0β > , if the curve ( )newg C intersects the curve ( )oldg C .
As follows from Eq. (4.8), the dependence ( )G R∆ can be monotonic
or nonmonotonic, depending on the magnitude of concentration gradi-
ent (Fig. 4.2).
Case (a) corresponds to full suppression of nucleation at sharp con-
centration gradients. Case (b) means a possibility of metastable nu-
cleus formation. Case (c) means the possibility of forming the stable
particle of new phase, size of which will increase with decreasing (with
time due to interdiffusion) concentration gradient.
Simple algebra gives the following expressions for the values
Fig. 4.2. Size dependence for Gibbs free energy change by nucleation of
spherical particles in the concentration gradient.
(a) ( )crit
1 5
C C
β β∇ > ∇ =
α γ
–nucleation forbidden;
(b) ( ) ( )crit crit
1 2
4
27
C C C
β β∇ > ∇ > ∇ =
α γ
–possibility of metastable nucleus;
(c) ( )crit
2
C C∇ < ∇ –nucleation possible.
452 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
( )crit
1,2
C∇ , corresponding to crossovers b a↔ and c b↔ :
( )crit
1 5
C
β β∇ =
α γ
, ( )crit
2
4
27
C
β β∇ =
α γ
. (4.10)
One can see the values of ( )crit
1
C∇ and ( )crit
2
C∇ are rather close, so that
the regime of metastable nucleation is difficult to detect. Moreover, we
shall see below that the shape optimisation excludes this regime (if one
does not consider stresses). Numerical estimates for the critical gradi-
ent for systems like Ni—Zr typically give ( )crit 8 1
1
10 mC −∇ ∝ .
4.1.2. Shape Optimisation
Evidently, since concentration gradient suppresses the nuclei growth in
longitudinal direction, nature will find possibilities to increase nucleus
volume (and decrease Gibbs free energy) by transversal growth. It means
that nuclei forming in the diffusion zone should be nonspherical. For
each fixed nucleus volume, one should take into account the shape opti-
misation. First attempt in this direction was made in 1991 [39]. Nuclei
(embryo) were supposed to be spheroids with symmetry (rotation) axis
along the C∇ -direction with parameters ( )|| ||R x and ( )R x⊥ ⊥ . In this
case, G∇ is a function of two arguments–volume V and shape parame-
ter ||R R⊥η = at fixed concentration gradient 1C L∇ = :
( )
22
||2 3 2
0|| || ||2 2 2
||
2 2
, 2
3 15 2 2
R RRg
G R R n g R R R R
L n R R
⊥⊥
⊥ ⊥ ⊥
⊥
′′ σ ∆ = π − ∆ + + + × −
2
|| || ||
2
|| ||
ln 1 , 1,
arcsin 1 , 1,
R R R
R R R
R R
R R
⊥ ⊥ ⊥
⊥ ⊥
− + > ×
− <
(4.11)
where
1
2
3
3
||
3
4
V
R
− = ⋅ η π
,
1
13
33
4
V
R⊥
= ⋅ η η
.
At every fixed volume V, an optimal shape ( )Vη is found by minimiz-
ing the function ( )|G V∆ η . The function ( )opt Vη increases to infinity
Nucleation and Growth in Nanosystems: Some New Concepts 453
at some value V
*
(Fig. 4.3), which is determined by the concentration
gradient (the larger is C∇ , the less is V
*). Dependence ( )( ), optG V V∆ η
looks different for large and for small concentration gradients (Fig. 4.4).
Fig. 4.3. Dependence of optimal shape,
||R R⊥η = , on nucleus volume.
Fig. 4.4. Dependence of Gibbs free energy on the volume of nucleus with opti-
mized shape: (a) ∇C > (∇C)crit; (b) ∇C < (∇C)crit. V0 = 3⋅10−29
m
3, ∆go = 10−20
J; σ =
= 0.5 J/m2; n = 1029
m
−3; g′′ = 10−16
J, critical width of diffusion zone appeared
to be L
crit
= 1/∇Ccrit
= 6.2⋅10−8
m, Vmax corresponds to some volume at which the
growing nucleus meets another laterally growing nucleus and stops the own
lateral growth.
454 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
Thus, the main result of previous subsection is repeated–the exis-
tence of concentration gradient, over which nucleation of intermediate
phase is forbidden. Yet, the possibility of metastable nuclei disap-
peared due to shape optimisation.
New results are: (1) the formation of plate-like nuclei (of course,
this result is not valid, if critical G∆ is much higher than 60kT) and
(2) decrease of nucleation barrier and corresponding increase of the
value of critical concentration gradient due to the shape optimisation.
Of course, suggestion of spheroidal shape is not rigorous. To verify
the validity of the above-mentioned results, we will present the direct
Monte Carlo simulation of the nucleus formation.
4.1.3. MC Simulation of Nucleus Shape in Concentration Gradient
We investigated the possible nucleation of the stable intermediate
phase in concentration gradient by means of Monte Carlo technique.
Let the concentration dependences of Gibbs potential for both phases
to be approximately parabolic with minima at 0 0 1/2new oldC C= = . Con-
centration profile in the parent phase in the vicinity of the forming
nucleus is approximated by linear dependence. In polymorphous mode,
nucleation proceeds fast in the frozen concentration gradient, and con-
centration changes start due to diffusion after nucleation. We divide
the homogeneous alloy into ‘elementary’ cells, each of them can trans-
form from old to new phase and vice versa, depending on thermody-
namic profit, which is determined by bulk driving force and the num-
ber of neighbouring cells in different phase states.
Simulation procedure. Each cell can exist in two-phase states–old and
new. The change of the phase state leads to change of bulk and surface
energy. For example, if the cell transforms from old to new state then
the change of Gibbs potential for the system is equal to
( ) ( )( ) ( )3 2new oldG g C g C a n Na∆ = − ⋅ + σ∆ . (4.12)
Here C is the concentration in the cell depending on its position (x–co-
ordinate, if the concentration gradient is parallel to x), n–atomic den-
sity, a–the cell size, σ–surface tension between old and new phases,
∆N is a change of number of neighbouring cells with different states
(even number from −6 to +6). If ∆G is negative, the transformation is
accepted, otherwise acceptance probability is found as exp( )G kT−∆
(Metropolis algorithm). To make the Monte Carlo procedure time sav-
ing, we try the state changes only for cells in the border layer of the
forming nucleus.
The following algorithm has been realized.
0) At first, all cells belong to old phase. Randomly we choose one cell
as a nucleation site and try its transformation according to Metropolis
Nucleation and Growth in Nanosystems: Some New Concepts 455
procedure ( 6N∆ = ).
1) One of the border cells is chosen randomly from the border set.
(Cell belongs to border set if it belongs to nucleus and has at least one
neighbour of old phase).
2) Cluster consisting of the chosen border cell and its six neighbours
is further considered. One of the seven cells of this cluster is chosen
randomly. This choice is accepted if the chosen cell is central (trans-
formation new→old) or if it is a neighbour belonging to old phase
(transformation old→new). Otherwise, attempt is repeated.
3) Change of the Gibbs potential for possible transformation is cal-
culated according to Eq. (4.12) and decision on acceptance/non-accep-
tance is made according to the Metropolis procedure.
Step (2) of the abovementioned algorithm artificially increases the
probability of nucleus growth. Otherwise, the subcritical embryo would
be most probably destroyed, and the formation of overcritical nucleus
would take very long computation time. The results of simulation are
presented for sharp and for small concentration gradients at Fig. 4.5.
a b
Fig. 4.5. Examples of nucleus shape simulation for sharp ((a)–∇С = 109
m−1) and small ((b)–∇С = 107 m−1) concentration gradients. Parameters of
simulation: σ = 0.15 J/m2, a = 1.5⋅10−10 m, n = 1029 m−3,
2
2
g
C
∂ ∆
∂
= 7.77⋅10−19 J,
g0
old − g0
new = 7.48⋅10−21 J.
456 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
4.2. Transversal Nucleation Mode
P. Desré and Yavari [32] first introduced this mode for cubic nuclei
without shape optimisation. In 1998, shape optimisation was done by
F. Hodaj, A. Gusak, and P. Desré [37] for the simplest case of paral-
lelepipeds. Let embryo (nucleus) in the form of parallelepiped
2 2 2h h r× × is born in the concentration gradient C∇ of metastable
parent phase (2r along C∇ ). Every thin slice 2 2h h dx× × with concen-
tration ( )newC x forms at the cost of slice dx∞ × ∞ × with concentration
( )oldC x according to the rule of parallel tangents (Fig. 4.7).
Driving force per atom of nucleus (not of total system) is equal to
( ) ( ) ( )old old new old new new
b
g
g g C C C g C
C
∂∆ = + − −
∂
(4.13)
Fig. 4.6. Scheme of transversal nucleation modes, vertical arrows show the
direction of redistribution fluxes of species in the slice dx.
Fig. 4.7. Rule of parallel tangents.
Nucleation and Growth in Nanosystems: Some New Concepts 457
and is determined by vertical interval between two parallel tangents as
shown on Fig. 4.7.
Using the rule of parallel tangents for each thin slice dx of paral-
lelepiped, one obtains:
( ) ( )( ) 2 2
1 24 2 4 4 4
C
C
x r
old new
x r
G n g C x C x h dx h hr
+
−
∆ = − ∆ → ⋅ + σ ⋅ + σ ⋅∫ , (4.14)
where ( ) ( )0oldC x C x C= + ⋅ ∇ , ( ) ( ) ( )( )0 0
old
new new old new
new
C x C C x C x
α= + −
α
,
σ1, σ2–surface tensions for phases, perpendicular and parallel to con-
centration gradient. Rather simple algebra, analogous to that in 4.1,
gives
( ) ( )22 2 3 2
1 28 8 2G h r C h r h hr∆ = −α ⋅ + γ ∇ + σ + σ , (4.15)
where
( )2
0 0
0 0
2 1
old new old
old new
old
new
C C
n g g
α − α = − + α− α
,
4
1
3
old old
new
n π α αγ = − α
. (4.16)
It is suitable to express G∆ as a function of volume
28V h r= and shape
parameter
h
r
ϕ = (
1 2
3 31
2
r V
−
= ϕ ,
1 1
3 31
2
h V= ϕ ):
( ) ( )2 5 2 1 23
1 3 3 3 32
1, 2 2
32
C
G V V V s V
−− ∇ γ
∆ ϕ = −α + ϕ + σ ϕ + ϕ
, (4.17)
where 2 1s = σ σ is a Wulf parameter.
Function ( )|G V∆ ϕ at every fixed volume has one minimum, which
is determined by condition 0G∂ ∂ϕ = , which gives an optimal shape:
( )22
12 4 32opt
opt
C Vh s s
r
γ ∇ ϕ = = + + σ
. (4.18)
For small volumes ( 0V → ), it leads to Wulf rule:
( ) 2 10opt V sϕ → = = σ σ ; (4.19)
for large volumes, the shape parameter increases as
1
2V :
458 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
( ) ( )
1
1 2
2 2
132opt V C V
γϕ → ∞ ≈ ∇ ⋅ ⋅ σ
(4.20)
that means the plate-like shape–concentration gradient limits the lon-
gitudinal size,
( )
( )
1
3 2
1 3
max 2
4
V
r r C
C
−
→∞
σ → = ∼ ∇
γ ∇
, (4.21)
but does not limit the transversal growth,
1
1 1 13
3 2 2
V
h V V V
→∞
∼ ⋅ ∼ → ∞
. (4.22)
The shape parameter depends on the product ( )2
C V∇ that means some
kind of scale invariance.
Thus, the volume dependence of G∆ for optimized shapes is follow-
ing:
( ) ( ) ( )
4
1 3
2 2 2 52
1 3
132 2 4 32
C C Vs s
G V V V
−
γ ∇ γ ∇
∆ = −α + + + + σ
( ) ( )
2 1
1 13 3
2 22 2 22 2
3
1
1 1
2 2
2 4 32 2 4 32
C V C Vs s s s
s V
−
γ ∇ γ ∇ + σ + + + + + σ σ
. (4.23)
One can see that, depending on the value of concentration gradient,
G∆ can be monotonously increasing or nonmonotonic with a maximum
(nucleation barrier):
a) ( )
1
2crit
1
4 2
3 3
C C
α α∇ > ∇ = σ γ
–nucleation forbidden,
b) ( )crit
C C∇ < ∇ –nucleation possible.
Thus, transversal mode, under condition of shape optimization, gives
qualitatively the same results as polymorphic mode:
1) nuclei should be more plate-like, the larger is the volume and the
larger is the concentration gradient, shape being determined by they
product ( )2
C V∇ ;
Nucleation and Growth in Nanosystems: Some New Concepts 459
2) nucleation is forbidden if concentration gradient exceeds certain
critical value, which can be about ( )crit 8 9 1
1
10 10 mC −∇ = − .
4.3. Total Mixing Mode of Nucleation
Another possibility of nucleation in the fixed gradient is a redistribu-
tion of components only inside the forming nucleus resulting in con-
stant concentration
newC and a new lattice, and unchanged gradient
outside nucleus. In this case, change of Gibbs free energy due to forma-
tion of nucleus is:
( ) ( )( ) ( )( )( )2 2
1 24 2 4 4
C
C
x r
new new old old
x r
G h hr n h g C x g C x dx
+
−
∆ = σ + σ + −∫ , (4.24)
Similar to previous subsection one obtains:
( ) ( )22 2 2 3
1 2 ||4 2 4 8G h hr h r C h r∆ = σ + σ − α + γ ∇ , (4.25)
||
4
3
oldnγ = − α . (4.26)
The main peculiarity here is the negative sign of ||γ . It means that, con-
trary to polymorphous and transversal modes, in the case of operating
total mixing mode the concentration gradient helps the nucleation.
Therefore, at any concentration gradient, a nucleation via total mixing
mode is always possible in thermodynamic sense. As we will see below,
kinetics can nevertheless suppress such a nucleation. Dependence
( )G∆ ϕ at fixed volumes is nonmonotonic with one minimum and one
maximum for small volumes (Fig. 4.8, a) and monotonously increasing
Fig. 4.8. ∆G versus shape dependence for total mixing mode at small and
large V(∇C)2.
460 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
for large volumes (Fig. 4.8, b).
Condition of extremum, 0G∂ ∂ϕ = , leads to the following equation
( )2
||2
1
0
32
C V
S
γ ∇
ϕ − ϕ + =
σ with two solutions
( )2
2
||
1,2
12 4 32
C VS S γ ∇
ϕ = ± −
σ
,
first of which corresponds to metastable minimum and second–to
maximum.
These solutions disappear, when ( )
2
2 1
||
8S
C V
σ∇ >
γ
, so that nucleus
should rapidly transform into needle (see Fig. 4.9). Thus, one has kind
of shape phase transition.
Obviously, all above-mentioned considerations are valid only for
constC∇ = , so that needle cannot exceed the size of diffusion zone.
Of course, total mixing mode should operate if the redistribution in
transversal direction in the parent phase is absent. For this diffusivity
of the new phase should be much larger than the diffusivity of old one.
In reality, all nucleation modes operate simultaneously. Description of
their interference should be made within the kinetic approach and will
be briefly reviewed in the next section.
5. KINETICS OF NUCLEATION. FOKKER—PLANCK APPROACH
Kinetics of nucleation is usually described by Fokker—Planck (FP)
equation for the size distribution of embryo/nuclei ( , )f t N , where f is a
number of clusters (say, per unit volume) containing N monomers
(with size
1/3( )R N∝ Ω ). In FP approach, the clusters are growing or
shrinking due to attachment or detachment of individual atoms, so
that movement of each cluster in the N-space (size space) is quasi-
continuous, step-by-step: clusters are not colliding with immediate
Fig. 4.9. ‘Phase shape transition’ for total mixing mode.
Nucleation and Growth in Nanosystems: Some New Concepts 461
coalescence and not splitting. Therefore, size distribution satisfies the
continuity equation in the size space:
,
f j
t N
∂ ∂= −
∂ ∂
(5.1)
where flux ( )j N of clusters in the size space contains both stochastic
and drift terms:
f
j f
N
∂= −ν + ∆ν ⋅
∂
. (5.2)
Stochastic term
f
N
∂−ν
∂ is due to randomly attaching and detaching at-
oms with average frequency ν . Drift term f∆ν ⋅ is determined by
thermodynamic profit of attachment comparing with detachment (or
vice versa):
G
N+ −
∂∆ν = ν − ν ∝ −
∂
.
We just modified this well-known approach by including the gradi-
ent term ( )2 5/3C Nγ ∇ into Gibbs energy change and taking the time de-
pendence of concentration gradient into account. Detailed derivations
can be found in [38]. Here we present the most characteristic figures.
5.1. Kinetics of the Intermediate Phase Nucleation
in the Concentration Gradient: Polymorphic Mode
Basic kinetic equation for the distribution function ( ),f N t for the
clusters of new phase:
( ) ( ) ( )
2
2
,f N t j
f f
t N N N
∂ ∂ ∂ ∂= − ∆ν ⋅ + ν ⋅ = −
∂ ∂ ∂ ∂
, (5.3)
Here
2
+ −ν + νν = is taken to be a constant value,
1 G
kT N
∂∆ν = − ⋅ ν
∂
(5.4)
In the case of polymorphous mode, the gradient term is positive (con-
centration gradient hinders the nucleation). Under assumption of
spherical nucleus shape, the dependence of Gibbs free energy on the
number of atoms in the nucleus has the following form:
( ) ( )
2 2
5 2
3 32 3 33 3
4
4 10 4m
g
G N g N C N N
n n
′′ ∆ = ∆ ⋅ + ∇ ⋅ + πσ ⋅ π π
(5.5)
462 A. M. Gusak, A. O. Bogatyrev, A. O. Kovalchuk, S. V. Kornienko et al.
Here g′′ is the difference of second derivatives of Gibbs free energy per
atom on concentration between the new phase and parent solution. If
the concentration gradient in the nucleation place changes according
to parabolic law:
( )2 1
4 parent
C
D t
∇ =
π
, (5.6)
then a drift term in FP-equation explicitly depends on time, which
physically means the lowering of nucleation barrier due to interdiffu-
sion in the parent phase.
It is convenient to use further the non-dimensional variables:
tτ ≡ ν , 0mg
kT
∆α ≡ < , 3 24 parent
g
kTC
D
ν′′
β ≡
π
, 3
8
3
C
kT
πσγ ≡ ,
2
3
3
3
4
C
n
= π
. (5.7)
Then
( )
2
2 3
12
3
,f N t f N
f
N N
N
∂ ∂ ∂ γ = + ⋅ α + β + ∂τ ∂ ∂ τ
. (5.8)
Numeric solution of this last equation has been obtained for the fixed
total number of nucleation sites (heterogeneous nucleation):
( ), constf N t dN =∫ . (5.9)
The evolution in time of the size distribution f(N) corresponds to evo-
lution of the potential field ∆G(N) but is shifted in time: For small an-
nealing times, when the concentration gradient remains sharp enough,
dependence ∆G(N) is monotonously increasing so that nucleation is
thermodynamically forbidden.
During this period, a distribution f(N) remains monotonously de-
creasing. After certain incubation thermodynamic ‘time’, when the
concentration gradient in the parent phase becomes less than critical
value, nucleation becomes thermodynamically possible (∆G(N) non-
monotonic with a maximum corresponding to nucleation barrier). Yet,
a distribution function f(N) reveals maximum not at once, but after
certain ‘kinetic incubation period’ (Fig. 5.1). We define an incubation
time as a period of peak formation for size distribution f(N) (not count-
ing an initial peak at minN N= ). Obviously, in dimensionless scale τ ,
the incubation time ( incτ ) should depend on the ratio of two kinetic pa-
rameters: ν and parentD . Dependences ( )inc parentDτ ν for different sur-
|
| id | nasplib_isofts_kiev_ua-123456789-133321 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1608-1021 |
| language | English |
| last_indexed | 2025-12-01T08:11:36Z |
| publishDate | 2004 |
| publisher | Інститут металофізики ім. Г.В. Курдюмова НАН України |
| record_format | dspace |
| spelling | Gusak, A.M. Bogatyrev, A.O. Kovalchuk, A.O. Kornienko, S.V. Lucenko, Gr.V. Lyashenko, Yu.A. Shirinyan, A.S. Zaporoghets, T.V. 2018-05-23T17:37:37Z 2018-05-23T17:37:37Z 2004 Nucleation and Growth in Nanosystems: Some New Concepts / A.M. Gusak, A.O. Bogatyrev, A.O. Kovalchuk, S.V. Kornienko, Gr.V. Lucenko, Yu.A. Lyashenko, A.S. Shirinyan, T.V. Zaporoghets // Успехи физики металлов. — 2004. — Т. 5, № 4. — С. 433-502. — Бібліогр.: 92 назв. — англ. 1608-1021 PACS: 05.70.Np, 64.75.+g, 66.30.Ny, 66.30.Pa, 68.55.Ac, 82.60.Nh, 82.60.Qr DOI: https://doi.org/10.15407/ufm.05.04.433 https://nasplib.isofts.kiev.ua/handle/123456789/133321 Review covers some new concepts in theory and modelling of the initial stages of solid-state reactions in thin films, multilayers, nanoparticles, and bulk nanocrystalline materials. The following topics are included: possibility of oscillatory ordering and nucleation in the sharp concentration gradient; criteria of suppression/growth of stable and metastable phases at the nucleation stage; possible nucleation modes in sharp concentration gradient and their competition; competitive nucleation and decomposition in small volumes; criteria of unambiguous choosing the parameters of discontinuous precipitation based on the balance and maximum production of the entropy; formation of nanostructure under uniaxial compression of single-crystalline alloy. Обзор содержит набор некоторых новых концепций в теории и моделировании начальных стадий твердофазных реакций в тонких пленках, мультислоях, наночастицах и в объеме нанокристаллических материалов. Обсуждаются следующие темы: возможность осцилляционного упорядочения и зародышеобразования в поле градиента концентрации; критерий подавления/роста стабильных и метастабильных фаз на стадии зародышеобразования; возможность различных мод зародышеобразования в поле градиента концентрации и их конкуренция; конкурентное зародышеобразование и распад в малых объемах; критерий однозначного выбора параметров ячеистого распада, основанный на балансе и максимуме производства энтропии; формирование наноструктур под воздействием одноосного сжатия монокристаллического сплава Огляд містить набір деяких нових концепцій в теорії і моделюванні початкових стадій твердофазних реакцій у тонких плівках, мультишарах, наночастинках та в об’ємі нанокристалічних матеріалів. Обговорено наступні теми: можливість осциляційного впорядкування і зародкоутворення в полі градієнта концентрації; критерій пригнічення/росту стабільних і метастабільних фаз на стадії зародкоутворення; можливість різних мод зародкоутворення в полі градієнта концентрації та їх конкуренція; конкурентне зародкоутворення і розпад в малих об’ємах; критерій однозначного вибору параметрів коміркового розпаду, що базується на балансі та максимумі виробництва Present work is supported by International Association for Promotion of Cooperation with Scientists from New Independent States of the former Soviet Union (INTAS grant #784, young NIS Scientist Fellowships Programme 2003–INTAS Ref. No. 03-55-1169), in part by CRDF (grant #UE1-2523-CK-03) and Ministry of Education and Science of Ukraine. Additionally to colleagues, acknowledged in Introduction, authors are grateful to F. d’Heurle, P. Gas, G. Martin, and J. Schmelzer for helpful discussions of nucleation/growth issues. en Інститут металофізики ім. Г.В. Курдюмова НАН України Успехи физики металлов Nucleation and Growth in Nanosystems: Some New Concepts Зародышеобразование и рост в наносистемах: некоторые новые концепции Зародкоутворення та ріст у наносистемах: деякі нові концепції Article published earlier |
| spellingShingle | Nucleation and Growth in Nanosystems: Some New Concepts Gusak, A.M. Bogatyrev, A.O. Kovalchuk, A.O. Kornienko, S.V. Lucenko, Gr.V. Lyashenko, Yu.A. Shirinyan, A.S. Zaporoghets, T.V. |
| title | Nucleation and Growth in Nanosystems: Some New Concepts |
| title_alt | Зародышеобразование и рост в наносистемах: некоторые новые концепции Зародкоутворення та ріст у наносистемах: деякі нові концепції |
| title_full | Nucleation and Growth in Nanosystems: Some New Concepts |
| title_fullStr | Nucleation and Growth in Nanosystems: Some New Concepts |
| title_full_unstemmed | Nucleation and Growth in Nanosystems: Some New Concepts |
| title_short | Nucleation and Growth in Nanosystems: Some New Concepts |
| title_sort | nucleation and growth in nanosystems: some new concepts |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/133321 |
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