# Data-driven analysis of grain-growth kinetics in duplex and triplex systems

P G Kubendran Amos111The authors contributed equally<EMAIL_ADDRESS>Ramanathan Perumal222The authors contributed equally. Arnd Koeppe  Britta
Nestler  Department of Metallurgical and Materials Engineering,
National Institute of Technology, Tiruchirappalli 620015, Tamil Nadu, India
Institute of Applied Materials (IAM-CMS), Karlsruhe Institute of Technology
(KIT),
Strasse am Forum 7, 76131 Karlsruhe, Germany  Institute of Digital Materials
Science (IDM), Karlsruhe University of Applied Sciences,
Moltkestr. 30, 76133 Karlsruhe, Germany

###### Abstract

Grain growth in multiphase polycrystalline systems are conventionally
represented, and discussed, by considering the temporal change in the average
radius of the constituent phase-grains, and overall microstructure,
separately. Despite this elucidation of grain growth, a convincing
understanding on how the evolution kinetics of an individual phase-grains
relates to the growth rate of entire microstructure is yet to be achieved.
Therefore, in this work, comfortably comprehensible statistical tools are
employed to realise the influence of the evolving phase-grains on the overall
grain-growth kinetics of multiphase systems, both duplex and triplex, with
varying phase-fractions. A dataset capable of rendering statistically
significant, and accessible, information is built by modelling, and analysing,
grain growth in twenty different polycrystalline systems, comprising of
fourteen triplex and five duplex microstructures with varying phase-fractions,
and one homogeneous single-phase system, in multiphase-field framework.
Statistical analyses of these wide-range of microstructures unravel that,
during isotropic grain-growth, the overall evolution kinetics of the duplex
and triplex systems are principally governed by the growth-rate of the major-
phase grains, irrespective of the phase-fractions. In other words, even though
introduction of a second phase in a single-phase polycrystalline system
considerably reduces the grain-growth rate of the resulting duplex
microstructure, the corresponding kinetics of the evolution is predominantly
dictated by the growth rate of the major-phase grains with relatively large
volume-fraction. Furthermore, through multiple non-linear regression, it is
realised that with increase in the volume-fraction of the major-phase grains,
the overall isotropic grain-growth rate exhibited by duplex and triplex
microstructures increase in accordance with a definite exponential relation.
Additionally, the present investigation unravels that, while the evolution of
individual phase-grains during grain growth in triplex microstructures exhibit
a level of interdependency, the least interaction is observed between the
grains of minor-phases with relatively low volume-fraction.

###### keywords:

Grain growth, coarsening, duplex microstructure, triplex microstructure,
kinetics, phase fraction

††journal: arXiv.org

##  1 Introduction

Microstructure of materials are meticulously engineered to cater the ever
progressing and more demanding needs. Properties, once thought to be
irreconcilable, are increasingly combined through appropriate combination of
phases in the microstructures [1, 2, 3]. Even though microstructures
characterised by more than one chemically-distinct phases have been around for
a considerable period of time, the distribution of the constituent phases
engenders a unique category of multiphase materials. For instance, in
polycrystalline pearlitic steel, the combination of ferrite and cementite, in
a specific phase-fraction, is observed in all grains, despite being separated
by the interfaces (grain boundaries) [4, 5]. Such grains can be treated as
chemically homogeneous, despite the presence of two distinct phases. However,
on the other hand, there are specialised steels, wherein the grains are not
chemically homogeneous but are exclusively associated with one of the
constituent phases [6]. In other words, instead of martensite and ferrite co-
existing in all grains of the polycrystalline microstructure, individual
grains assume a phase, thereby establishing a chemical inhomogeneity in the
system [7]. Polycrystalline materials, characterised by these chemically-
distinct grains, are qualified contextually as multiphase, and depending on
number of constituent phases, these materials are referred to duplex, triplex
and such.

Besides steel, multiphase microstructures are established in wide-range of
materials to achieve desired combination of properties [8, 9]. The extensive
applicability of two- and three-phase titanium alloys is primarily due to
properties which is a direct consequence of their multiphase microstructure.
The prevalence of multiphase polycrystalline arrangement in high-entropy
alloys is a principal reason for their characteristic behaviour [10, 11]. In
ceramics, mechanical properties including fracture toughness are noticeably
enhanced by two-phase microstructure. Duplex system of alumina and silicon
carbide is a prime example of such behaviour-enhanced ceramics [12, 13].
Moreover, it has been reported that the introduction of an additional phase,
which institutes a triplex microstructure, exacts a more preferred response
from ceramics to an imposed mechanical conditions [14, 15]. Similar favourable
effects of multiphase microstructure are observed in composite materials as
well [16].

In addition to the characteristic feature of the phases associated with the
multiphase materials, the emerging properties are also considerably influenced
by the volume fraction of the constituent phases, $i.e$ phase-fraction, and
average grain size. Consequently, to achieve desired behaviour, multiphase
materials with varying phase-fraction ranging from minimal minor-phase (close
to 1%) to equi-fraction (50-50 in duplex), and appropriate grain sizes, are
fabricated [17, 18]. While material-specific processing techniques are
employed for introducing the necessary phase-fraction, the required grain
sizes are obtained through a rather well-known microstructural transformation
called grain growth [19]. The mechanism of grain growth in multiphase systems
is significantly different single-phase polycrystalline microstructure. In
polycrystalline materials with chemically-homogeneous grains, the local
diffusion of atoms across the grain boundaries, which consequently leads to
its migration, ultimately governs the grain growth. In other words, in single-
phase systems, the grain growth is primarily dictated by the movement of the
interface (or) grain boundaries. However, in multiphase microstructures, the
association of grains to a specific constituent phase adds an additional facet
to the conventional grain growth. Besides reducing the number of grains, grain
growth in multiphase systems conserves the characteristic volume-fraction of
the phases. Therefore, the curvature-driven transformation, which minimises
the overall grain-boundary energy, in multiphase materials is, in principle, a
combination of coarsening and grain growth [20]. Typifying features of both
coarsening and grain growth, which respectively are preservation of volume
fraction and growth of larger grains at the expense of smaller ones, are
simultaneously observed in multiphase materials. For this reason, the energy-
minimising curvature-driven transformation in multiphase systems is at times
referred to as concurrent grain growth and coarsening. Correspondingly, as
opposed to interface migration, the long-range diffusion of atoms dictates
grain growth in multiphase polycrystalline systems [21]. Regardless of the
mechanism, owing to the influence of the grain size on the properties rendered
by the materials, grain growth in multiphase systems have been extensively
analysed [22, 23]. Moreover, it is conceivable, and indeed reported that, a
change in the average grain-size of a multiphase material, while employed in
an application, results in undesired behaviour. In other words, besides the
direct effect of grain-size on the properties of multiphase materials, its
life in an application is dictated by the rate at which the grains grow.
Solid-oxide fuels with triplex microstructure are prime examples of this
relation between the growth kinetics and life of a material [24]. Considering
this influence of grain size, investigations have been geared towards
understanding the kinetics of grain growth in multiphase polycrystalline
system.

Experimental techniques generally pose definite practical difficulties when
adopted for comprehensive analyses of grain growth. Therefore, theoretical
approaches have long since been adopted to complement, and extend, the
existing understanding [25, 26, 27]. These theoretical studies, particularly
one involves multiphase-field models, have been offering critical insights on
mechanism and kinetics of grain growth in multiphase system, which are
consistent with the experimental observation [28, 29, 30]. Existing
investigations, and resulting understanding, of grain-growth kinetics in
multiphase polycrystalline systems can broadly be categorised into two. One
relates material-specific parameters, including diffusivities and grain-
boundary energy anisotropies, to the rate of grain growth [31, 32], while the
other focuses on the effect of microstructural features like phase-fractions
[18, 33]. While the material-specific studies are, in principle, inherently
bound by the choice parameter(s), the analyses involving phase-fraction offer
more generalised insights that are relevant to a wide-range of multiphase
polycrystalline systems.

In-keeping with the change in grain-growth mechanism, the investigations
focusing on microstructural features unravel that the introduction of second
phases significantly reduces the kinetics of overall evolution. Moreover, it
is realised that the expression capturing the temporal change in the average
radius of the polycrystalline system reflects coarsening kinetics,
$\bar{R}^{n}\propto t$ with $n=3$, irrespective of the volume-fraction of the
second phase [21, 34]. Even though the major and minor phase(s) - categorised
based on volume-fraction- , and the entire polycrystalline system as a whole,
evolve at a rate that complies with the coarsening power-law, the kinetic-
coefficient ($k$) varies with the phases, particularly their volumes [18]. For
instances, in a duplex microstructure characterised by unequal volume-fraction
of constituent phases, the grains of the major phase, with higher phase-
fraction, grow at the noticeably faster rate when compared to the minor phase.
This disparity in the kinetics of the grain growth between the phases is
indicated by the difference in the kinetic-coefficient ($k$). The relatively
sluggish growth of the minor phase(s) is attributed to the increased distance
between the respective chemically-similar grains, and complex diffusion-
pathways that facilitate the growth, when compared to the proximity of the
major phases. Furthermore, attempts have been quantify the effect of volume-
fraction on the kinetic-coefficients of major- and minor-phases by comparing
the evolution of the microstructures with varying phase-fractions [33].
Despite these apparent advancements, there continues to exist certain aspects
to the grain-growth kinetics that prevent a comprehensive understanding of the
evolution of multiphase polycrystalline systems.

In addition to the characteristic nature, and volume-fraction, of the phases,
the properties of multiphase systems depend on the overall grain-size of the
polycrystalline microstructure. However, studies aimed at explicating the
grain-growth rate in multiphase materials primarily report, and discuss, on
the difference in the kinetics exhibited by the major- and minor-phases, while
largely overlooking the evolution of the entire microstructure [35, 12].
Moreover, the collective influence of the growth rate of major- and minor-
phases on the temporal change in overall average size of the grains is yet to
be convincingly understood. Besides, in three (or more) phase systems, the
interdependency between the grain-growth rate of different phases have not
been thoughtfully considered so far. Amongst others, in the present work,
attempts are made to address the aforementioned questions on the grain-growth
kinetics of the overall multiphase microstructure through appropriate
statistical analysis.

##  2 Devising ‘multidimensional’ dataset

Persuasive understanding on how the evolution of grains of an individual phase
relates to the overall grain growth in a multiphase system, can hardly be
gained from analysing of a few microstructures. To that end, a
‘multidimensional’ dataset is developed by modelling grain growth exhibited by
duplex and triplex microstructures in the multiphase-field framework [36, 37].
Thermodynamical underpinnings of the adopted model is relatively well-
established, and is often acclaimed to be a computationally-efficient
alternate [38, 39]. Even though the complete formulation of the present
approach [40, 41], its ability to incorporate different modes of mass
transfer, and recover sharp-interface solutions is exhaustively discussed
elsewhere [42, 43], a brief outline focusing on the key aspects in rendered
here.

###  2.1 Multicomponent multiphase-field model

Multiphase-field approach is characterised by the introduction of scalar
variable(s) which distinguishes the different phases in the system.
Conventionally, while modelling polycrystalline microstructures, these
variables, called phase-field ($\phi(\bm{x},t)$), are employed to
differentiate various grains, and their spatio-temporal evolution translates
to grain growth. Across the grain boundary separating two grains, phase-field
assumes a value of $\phi=0$ in one grain, while the other is realised by a
constant non-zero value, which is generally $\phi=1$. The diffuse region,
wherein the value of the phase-field gradually varies,
$\phi(\bm{x},t)\in(0,1)$, describes the grain boundary. Given that, in most
cases, the grains in the polycrystalline systems are chemically homogeneous,
phase-field typifying the various grains are introduced as a tuple, which is
expressed as

$\displaystyle\mbox{{\bm{$\phi$}}}=\left\\{\phi_{1},\phi_{2},\dots,\phi_{N}\right\\},$
(1)

where $N$ denotes the total number of grains in the system. However, since the
present approach models grain growth in multiphase systems comprising of $N$
phases, wherein numerous grains are associated each phase, the corresponding
phase-field is extended and written as

$\displaystyle\mbox{{\bm{$\phi$}}}=\Big{\\{}\underbrace{\\{\phi_{\alpha}^{1},\phi_{\alpha}^{2}\dots\phi_{\alpha}^{q_{\alpha}}\\}}_{\bm{\phi_{\alpha}}},\underbrace{\\{\phi_{\beta}^{1},\phi_{\beta}^{2}\dots\phi_{\beta}^{q_{\beta}}\\}}_{\bm{\phi_{\beta}}}\dots\underbrace{\\{\phi_{N}^{1},\phi_{N}^{2}\dots\phi_{N}^{q_{N}}\\}}_{\bm{\phi_{N}}}\Big{\\}}.$
(2)

Moreover, in this formulation, number of grains sharing a given phase is
represented by $q_{i}$ where $i\in\\{\alpha,\beta,\dots,N\\}$. By assigning
appropriate concentration, grains of a given phase are distinguished from the
rest,
$\\{\mbox{{\bm{$\phi$}}}(\mbox{{\bm{$c$}}})=\mbox{{\bm{$\phi$}}}_{\alpha}(\mbox{{\bm{$c$}}})|\mbox{{\bm{$c$}}}=\bm{c}_{\alpha}\\}$.
Similar to phase-field, in order to encompass the $K$-different chemical
components, the concentration is introduced as a tuple. Correspondingly,
concentration of a random grain $m$ associated with phase-$\alpha$ reads

$\displaystyle\mbox{{\bm{$c$}}}^{\alpha}_{m}=\left\\{c^{\alpha}_{m:i},c^{\alpha}_{m:j},\dots,c^{\alpha}_{m:K}\right\\}.$
(3)

The homogeneity in the chemical composition of the grains associated with a
given phase, say $\alpha$, yields

$\displaystyle\mbox{{\bm{$c$}}}^{\alpha}_{1}=\dots=\mbox{{\bm{$c$}}}^{\alpha}_{m}=\dots=\mbox{{\bm{$c$}}}^{\alpha}_{q_{\alpha}}\equiv\mbox{{\bm{$c$}}}^{\alpha}=\left\\{c^{\alpha}_{i},c^{\alpha}_{j},\dots,c^{\alpha}_{k}\right\\}.$
(4)

To ensure that the volume-fraction of the phases remains unaltered despite the
evolution, equilibrium composition is assigned to the corresponding phases. In
other words, since the current approach attempts to model grain growth in
multiphase system, $\mbox{{\bm{$c$}}}^{\alpha}$ represents a tuple of
equilibrium composition that characterises phase-$\alpha$.

Following the conventional framework, the overall energy-density of the
multiphase polycrystalline system is formulated as the combination interface
(grain boundaries) and bulk (grain) contribution [44]. Correspondingly, by
incorporating the appropriate phase-field and concentration, the energy-
density of the system comprising of $N$-phases and $K$ chemical components is
written as

$\displaystyle\cal{F}(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})$
$\displaystyle=\cal{F}_{\text{int}}(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})+\cal{F}_{\text{bulk}}(\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})$
(5)
$\displaystyle=\int_{V}f_{\text{int}}(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})+f_{\text{bulk}}(\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})\mathrm{d}{V},$

where $f_{\text{bulk}}(\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})$ and
$f_{\text{int}}(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})$ are the
respective energy-contribution of grain and grain boundary with $V$
representing the volume. The interface contribution,
$f_{\text{int}}(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})$, akin
to most multiphase-field techniques, comprises of a gradient- and potential-
energy term. Grain-boundary energy densities, and corresponding anisotropies,
are introduced to the system through the interface-energy contribution [45].
While multi-well potentials are generally employed to penalise phase-field,
and ensure its bounds, in the present work, an obstacle-type potential
operating in combination with Gibbs simplex is involved [46]. Furthermore, the
energy contributions from the grains,
$f_{\text{bulk}}(\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})$, which is reasonably
assumed to be insignificant while modelling grain growth in single-phase
system, is formulated as the interpolation of the energy contribution of the
individual grains,
$f(\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})=\sum_{\alpha}^{N}\sum_{m}^{q_{\alpha}}f^{\alpha}_{m}(\mbox{{\bm{$c$}}}^{\alpha})h(\phi_{\alpha}^{m})$.
Given that the contribution of the individual grains is dictated by the
characteristic equilibrium-composition of the associated phases, the volume of
the phases during grain growth is preserved by the energy-density term,
$f_{\text{bulk}}(\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})$ [47, 48].

The spatio-temporal evolution of phase-field, which translates to grain
growth, is formulated by considering phenomenological minimisation of the
overall energy-density of the system. Correspondingly, the evolution of a
random grain $m$, which is associated with phase-$\alpha$, is dictated by

$\displaystyle\tau\varepsilon\frac{\partial\phi_{\alpha}^{m}}{\partial t}$
$\displaystyle=-\frac{\partial\cal{F}(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})}{\partial\phi_{\alpha}^{m}}$
(6) $\displaystyle=\varepsilon\left[\bm{\nabla}\cdot\frac{\partial
a(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})}{\partial\bm{\nabla}\phi_{\alpha}^{m}}-\frac{\partial
a(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})}{\partial\phi_{\alpha}^{m}}\right]-\frac{1}{\varepsilon}\left[\frac{\partial
w(\mbox{{\bm{$\phi$}}})}{\partial\phi_{\alpha}^{m}}\right]-\left[\frac{f^{\alpha}_{m}(\mbox{{\bm{$c$}}}^{\alpha},\phi_{\alpha}^{m})}{\partial\phi_{\alpha}^{m}}\right]-\Lambda,$

where the Lagrange multiplier $\Lambda$ is introduced to ensure that the
summation of phase-fields at any point in the system is $1$. Moreover, in the
above evolution equation, while
$a(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})$ and
$w(\mbox{{\bm{$\phi$}}})$ correspond to the gradient and potential energy
terms, the parameter dictating interface-width, and its stability during the
migration, are denoted by $\varepsilon$ and $\tau$, respectively.

In multiphase-field models, wherein the energy contribution of the bulk phases
are described based on the dependent-concentration, the corresponding driving-
force, which emerges from
$\frac{f^{\alpha}_{m}(\mbox{{\bm{$c$}}}^{\alpha},\phi_{\alpha}^{m})}{\partial\phi_{\alpha}^{m}}$
in Eqn. (6), and dictates the evolution of phase-field, can be viewed as the
difference in the Legendre transform of the free-energy densities. This
understanding forms the basis of the grand-potential approach, and when
consistently extended assumes chemical potential as the continuous and dynamic
variable replacing phase-dependent concentration [49]. Given its computational
efficiency, this approach is adopted in the present work, and the driving-
force dictating phase-field evolution is formulated by treating chemical
potential as the dynamic variable. The temporal evolution of the chemical
potential, which principally governs the bulk driving-force in phase-field
evolution, is written as

$\displaystyle\frac{\partial\mu_{i}}{\partial
t}=\left\\{\bm{\nabla}\cdot\left[\sum_{j=1}^{K-1}\bm{M}(\mbox{{\bm{$\phi$}}})\bm{\nabla}\mu_{j}\right]-\sum_{\alpha}^{N}\sum_{m}^{q}c_{i}^{\alpha}\frac{\partial\phi_{\alpha}^{m}}{\partial
t}\right\\}\left[\sum_{\alpha}^{N}\sum_{m}^{q}h(\phi_{\alpha}^{m})\frac{\partial
c_{m:i}^{\alpha}}{\partial\mu_{j}}\right]_{ij}^{-1},$ (7)

where $\mu_{i}$ denotes the continuous chemical-potential of component-$i$.
The mobility of the migrating elements, in the multicomponent setup, is
dictated by matrix $\bm{M}(\mbox{{\bm{$\phi$}}})$ which also facilitates the
incorporation surface diffusion [42, 43]. The phase-dependent concentration of
component-$i$ in random grain $m$ belonging to phase-$\alpha$, and the
corresponding interpolation function, are represented by $c_{m:i}^{\alpha}$
and $h(\phi_{\alpha}^{m})$, respectively. The evolution of the different
microstructures with varying phase-fractions, in this work, are modelled by
solving the Eqns. (6) and (7).

###  2.2 Simulation setup

Existing studies unravel that, unlike Zener pinning [50], the overall trend in
grain-growth kinetics exhibited by the individual phases, and entire
microstructure, of the multiphase system are largely independent of the
dimensionality of the simulation domain [18, 51, 33]. In other words, in both
two- and three-dimensional setup, similar disparity in the evolution kinetics
of major-, minor- and equifraction phases, in relation to overall growth rate,
has been reported. Therefore, in the present work, the grain growth in various
multiphase-polycrystalline systems are modelled in two-dimensional framework.
Moreover, irrespective of the variation in phase-fraction, two-dimensional
domains of similar configurations are adopted for all numerical studies.

Two-dimensional domains considered for the current investigations are
uniformly discretised into $2048\times 2048$ cells of identical dimension,
$\Delta$x = $\Delta$y = $5\times 10^{-7}$m, through the finite-difference
scheme. Polycrystalline microstructures comprising of approximately $10000$
grains are instituted over the discretised domain through Voronoi
tessellation. These grains are associated to the constituent phases by
assigning the characteristic chemical composition. Since the grains are
equiaxed with almost similar size, the required phase-fraction in the
microstructure is achieved by relating the appropriate number of randomly
distributed grains to the corresponding phases. In other words, duplex
microstructure comprising of 33$\%$ minor phase is devised, in the initial
stages, by assigning the respective chemical composition to the one-third of
the grains randomly.

---
Figure 1:  A simplex, analogous to ternary isotherm, depicting all-possible
phase-fractions in duplex and triplex microstructures, along with single-phase
systems (vertices). The section that encompasses the varied microstructures
considered in this work is distinguished, and each point referring to a
specific system is highlighted and designated as $s1$, $s2$, and such.

Given that the principal focus of this work is to understand the evolution
kinetics of different phases in relation to each other, and to the overall
grain-growth rate exhibited by the entire microstructure, a rather
straightforward distinction is made between the phases. While a binary system
with two chemical component, $\tilde{i}$ and $\tilde{j}$, is considered for
establishing duplex microstructure, three-phase microstructure is construed in
the framework of ternary system with components $\tilde{i}$, $\tilde{j}$ and
$\tilde{k}$. In duplex system, $\alpha$-phase is a $\tilde{i}$-rich phase with
equilibrium composition of $c^{\alpha}_{\tilde{i}:eq}=0.9$, and
$c^{\alpha}_{\tilde{j}:eq}=0.9$ characterises matrix $\gamma$-phase, wherein
both concentrations are expressed in mole fraction. Concentration of the
solvent in $\alpha$\- and $\gamma$-phase remains unaltered in the triplex
system, while the remnant content is equally partitioned between solutes,
$\\{\tilde{j},\tilde{k}\\}$ and $\\{\tilde{i},\tilde{k}\\}$, respectively. The
$\beta$-phase, exclusively introduced in the three-phase systems, is
characterised by composition
$c^{\beta}_{\tilde{i}:eq}=c^{\beta}_{\tilde{j}:eq}=0.05$. Since the present
investigation is primarily interested in microstructural features like phase-
fractions, diffusivities of the components are assumed to be identical, and
unit matrix is correspondingly incorporated in the formulation. Moreover, the
energy-densities of the grain boundaries, irrespective of the chemical-
composition of the grains they separate, is treated as isotropic and unity.
The length scale parameter, $\varepsilon$, is appropriately defined such that
the diffuse interface is of constant thickness comprising of four cells [52].

The temporal evolution of the dynamic variables, phase-field and chemical
potential, that dictate the microstructural changes in the multiphase
polycrystalline system, are solved over the homogeneous cells of the two-
dimensional domain by forward-marching Euler’s scheme. In order to ensure that
the computational resources are optimally used, the domain is decomposed into
smaller segments, and dealt simultaneously, through Message Passing Interface
(MPI). The complexity of the numerical treatment is reduced by suitably non-
dimensionalising the input parameters, and incorporating them as dimensionless
values [42].

###  2.3 Varying phase-fractions

---
Figure 2:  Microstructure corresponding to each point in the section of the
simplex that includes homogeneous systems along with duplex and triplex
microstructures with varying volume-fraction of the constituent phases.

In applications, depending on the material need, multiphase systems with
varying degree of phase-fractions are employed. Though investigating every
combination of the phase-fraction would be redundant, convincing level of
understanding can only be gained by considering relatively increased number of
phase-fractions. Particularly, since the present study adopts statistical
techniques to explicate the kinetic relation between the evolving phases, the
accuracy of the outcome depends on the wealth of information (data) available.
To that end, in this work, grain growth in twenty different systems, which
encompasses one homogeneous, five duplex and fourteen triplex microstructures,
is modelled, and ‘multidimensional’ dataset is built by monitoring the
temporal evolution of the grains.

As opposed to random consideration of different volume-fractions of phases, a
systematic choice of various phase-fraction is made from a 2-simplex. The
simplex, in its entirety, along with the section focused for the current study
is shown in Fig. 1. The points within, and on, the 2-simplex can be
interpreted in a manner akin to the ternary isotherm. Correspondingly, while
the three vertices indicate the homogeneous microstructure of phase-$\alpha$,
-$\beta$ and -$\gamma$, the duplex microstructures are encapsulated by the
edges joining the vertices. Any point within the simplex represents triplex
system, with phase-fraction dictated by its position. As illustrated in Fig.
1, a section of the simplex emanating from the vertex characterising the
homogeneous $\gamma$-microstructure is considered for the present analyses.
This section of the simplex renders a wide-variety of polycrystalline systems
ranging from single phase homogeneous to triplex with equifraction of
constituent phases. Moreover, owing to the configuration of the section in
Fig. 1, $\gamma$ -phase acts as the matrix for the duplex and triplex
microstructures with unequal volume-fractions of phases. Multiphase
microstructures corresponding to the different points of the simplex-section
is collectively illustrated in Fig. 2. Though the volume-fraction of the
minor-phases can be as low as 5$\%$, the grains associated with these phases
hardly occupy a position on the grain boundary. In other words, despite the
low volume-fraction and reduced size, the grains of the minor phases seldom
render an influence analogous to the particles in Zener pinning. Moreover, the
reduced size of the grains associated with minor phases, in the initial stages
of the grain growth, is consistent with experimental observations [23, 53, 54]
and existing theoretical studies [18, 33].

##  3 Results and discussion

---
Figure 3:  Temporal change in the average radius of the phase-$\alpha$ and
$-\gamma$ grains, $\bar{R}_{\alpha}(t)$ and $\bar{R}_{\gamma}(t)$, and the
entire microstructure, $\bar{R}(t)$, for homogeneous, and duplex systems
characterised by $50\%$ and $10\%$ of phase-$\alpha$.

By monitoring the grain growth exhibited by the homogeneous and multiphase
systems, a ‘multidimensional’ dataset is devised which essentially comprises
of temporal change in the average radius of the individual phase-associated
grains, and the entire microstructure as a whole. (A dataset can truly be
multidimensional only when its dependent variable is governed by more than one
independent variable. Considering that explicating the multivariate nature of
the overall grain-growth kinetics in multiphase system is the ultimate outcome
of the current study, the term ‘multidimensional’ is written within quotes.)
This dataset is analysed through comfortably realised statistical techniques
to unravel the effect of the individual phases on the evolution kinetics of
the entire system. Since the present study focuses primarily on the kinetics
of the grain growth, the topological changes and the distribution of the
grains are largely overlooked.

###  3.1 Duplex microstructures

Duplex microstructure comprises of two distinct phases, and is characterised
by grains associated with one of these constituent phases, $\alpha$ and
$\gamma$. Despite the inhomogeneity in the concentration distribution, in
duplex microstructures, given that the grain growth occurs in a continuum, the
temporal evolution of one phase, and its corresponding kinetics, is inherently
coupled with the other. This is illustrated and discussed in Appendix. In
other words, the growth rate exhibited by the duplex microstructures with
varying phase-fractions can be convincingly expressed by considering kinetics
of the only one of the evolving phases. Therefore, the aim of the present
investigation in duplex microstructure reduces to identifying which of the
phases, major or minor, principally governs the kinetics of overall grain-
growth.

####  3.1.1 Comparison with single phase microstructure

Before proceeding to realise the degree of influence rendered by the different
phases on the overall growth-kinetics exhibited by the duplex microstructure,
rather straightforward investigations are pursued to verify the outcomes of
the present approach in relation to the existing reports [18, 21]. Even
though, the outcomes of the modelling technique in relation to the established
theories and observations have already been reported elsewhere [42], certain
relevant aspects are analysed in a statistical framework here.

In Fig. 3, the progressive change in the average radius of homogeneous and two
duplex microstructures with time are presented. Moreover, the temporal
increase in the average radius of the phase-associated grains are monitored,
and included in this illustration. While $\bar{R}(t)$ represents the average
radius of the entire polycrystalline microstructure, the corresponding
parameter for the grains of phase-$\alpha$ and -$\gamma$ in duplex
microstructures is respectively denoted by $\bar{R}_{\alpha}(t)$ and
$\bar{R}_{\gamma}(t)$. Moreover, the multiphase microstructures in this, and
subsequent, discussions are described based on the volume fraction of the
minor phase. For instance, $\alpha 10$ indicates duplex microstructure with
$10\%$ phase-$\alpha$, while equifraction system are denoted $\alpha 50$.

Irrespective of the nature of the microstructure, homogeneous or otherwise,
Fig. 3 shows a continual increase in the average radius reflecting the grain
growth exhibited by the system. With the introduction of a second-phase in the
microstructure, a significant decrease is observed in the rate at which the
radius increases with time. This noticeable change in the kinetics is
predominantly due to the change in grain-growth mechanism, which is governed
by the long-range diffusion of the chemical components in duplex
microstructure. Moreover, in system with equal volume-fraction of phases, the
temporal increase in average radius of the entire microstructure and
individual phases are largely identical with marginal deviation. On the other
hand, in $\alpha 10$, significant disparity is noticed in the rate at which
the major-phase grains evolve when compared to the minor-phase. Fig. 3
illustrates that the increase in the average radius of the entire duplex
microstructure lies in between the growth exhibited by the individual phases.
The difference on the growth kinetics between the phases, exclusively in the
duplex microstructure characterised by unequal volume-fraction, is due to the
corresponding distribution of the phases. Owing to its reduced volume, the
grains of the minor phase in $\alpha 10$ microstructure are considerably
separated when compared to major-phase grains. Therefore, the diffusion path,
which the chemical components need to transverse to achieve grain growth, is
longer, and more convoluted. Consequently, the growth rate exhibited by the
minor-phase grains is significantly lower the grains of phase-$\gamma$. So
far, no convincing argument has been made on how the evolution kinetics of the
entire duplex microstructure relates to the different growth-rates adhered-to
by the major- and minor-phase grains, and to that end, this becomes the
primary focus of the current analysis.

---
Figure 4:  Correlation coefficient characterising the proportionality between
time and the average radius of individual phase-grains and entire
microstructure, raised to different exponents, for homogeneous and duplex
systems $\alpha 50$ and $\alpha 10$.

####  3.1.2 Grain growth kinetics

Even though the microstructures illustrated in Fig. 3 render a progressive
increase in the average radius with time, owing to the difference in the
governing mechanism, the exponent of the power law capturing the growth
kinetics vary depending on the nature of the system [18, 21]. While the
exponent $n=2$ characterises grain growth in homogeneous system, evolution
kinetics of the individual phases, and the duplex microstructure as a whole,
largely follow relation $\bar{R}^{3}(t)\propto t$. In order to affirm that the
evolution of the entire microstructure, and its corresponding phases, adhere
to the power law, the temporally varying average radius is raised to different
exponents, $n=\\{1,2,3,4\\}$, and related to time. The correlation coefficient
(Pearson) characterising the relation between the average radius with the
various exponent and time is ascertained, and graphically represented in Fig.
4. In this illustration, $<\bar{R}^{n},t>$ denotes the correlation coefficient
between the average radius raised to order n and time. Fig. 4 shows that, in
homogeneous system ( $\alpha 0$), maximum correlation is observed when $n=2$,
thereby indicating that the grain growth in this microstructure adheres to the
power law, $\bar{R}^{2}(t)\propto t$. Furthermore, correlation coefficient
relating the average radius of the individual phases, and entire duplex
microstructures, with time, is higher when $n=3$, which implies that the
evolution in the multiphase systems complies to the established power law
[21]. Even though it might appear that, in Fig. 4, for equifraction duplex
system ($\alpha 50$) , the maximum correlation is observed in $n=4$. However,
given the marginal difference when compared to $n=3$, such consideration leads
to ‘overfitting’ , thus returning to $\bar{R}^{3}(t)\propto t$ as the
statistically sound relation.

####  3.1.3 Ascertaining governing factor

As shown in Fig. 3, the change in the average radius of the entire
microstructure with time, in duplex systems with unequal volume-fraction of
phases, invariably lies between curves representing the increase in the radius
of the individual phase-grains. Moreover, existing works unravel that, in
radius versus time plot, the relative position of the overall microstructure
curve varies with respect to the phase-associated grains curves, as the phase-
fraction changes [18, 33]. While the effect of phase-fraction on the evolution
of the individual phases are convincingly elucidated, its influences on the
growth rate of entire duplex microstructure is yet to be sufficiently
addressed. In order to identify this influence, more particularly, to answer
the resulting question on evolution kinetics of which phase, major or minor,
principally effects the growth rate of entire duplex microstructure, each
system is statistically analysed. Statistical programming language R is
employed for these, and all other relevant, investigations in this work.

---
Figure 5:  Coefficient of determination, quantifying the effect of growth
kinetics of individual phase-grains on the overall grain-growth rate of the
entire system, is estimated using Eqn. (8) for different duplex microstructure
with varying phase-fraction.

From the dataset comprising of temporally varying average radius, the growth
rate exhibited by the individual phase-grains ($d\bar{R}_{\alpha}/dt$ or
$d\bar{R}_{\gamma}/dt$) and entire microstructure ($d\bar{R}/dt$), at every
instance $(t)$, is determined for each duplex system. Subsequently, by
treating the growth rate of the individual phase-grains, and entire
microstructure, as response and predictor variable, respectively, the
corresponding kinetics are related to the each other. From the emerging
relation, the coefficient of determination, $\chi$, for the combination of an
individual phase-grains and overall microstructure is estimated. Scatter plots
that illustrates the dependency of $d\bar{R}_{\alpha}/dt$ or
$d\bar{R}_{\gamma}/dt$ and $d\bar{R}/dt$ are included in Appendix2.

For a given duplex system, following the conventional description, the
coefficient of deterioration considering the growth kinetics of minor
phase-$\alpha$ ($d\bar{R}_{\alpha}/dt$) and entire microstructure
($d\bar{R}/dt$) is calculated by

$\displaystyle\chi_{\alpha}=\frac{\chi^{\text{SST}}_{\alpha}-\chi^{\text{SSE}}_{\alpha}}{\chi^{\text{SST}}_{\alpha}},$
(8)

where $\chi^{\text{SST}}_{\alpha}$ is estimated by treating the instantaneous
growth-rate of overall microstructure as univariate parameter, and summing-up
the squares of the disparity (error) between the individual values and the
mean. On the other hand, $\chi^{\text{SSE}}_{\alpha}$ represents the sum of
the squared differences between the datapoints and regression line relating
the instantaneous kinetics of the $\alpha$-phase grains and overall duplex-
microstructure. Based on the description of the coefficient of determination
in Eqn. (8), $\chi_{\alpha}$ can be viewed as a parameter that quantifies the
effect of $\alpha$-grains growth-kinetics on the evolution rate of entire
microstructure. Therefore, in addition to $\chi_{\alpha}$, the corresponding
parameter that realises the influence of the major-phase growth-kinetics
($d\bar{R}_{\gamma}/dt$) on the overall evolution rate, $\chi_{\gamma}$ is
appropriately determined for all the different duplex microstructures
considered in this investigation.

Coefficients of determination separately quantifying the role of
$d\bar{R}_{\alpha}/dt$ and $d\bar{R}_{\gamma}/dt$ in overall growth-rate
exhibited by the microstructure, $\chi_{\alpha}$ and $\chi_{\gamma}$, is
calculated for different duplex systems with varying phase-fractions and
plotted in Fig. 5. The variation observed in the coefficients of
determination, across the different duplex systems, unravels that the effect
of individual phase-grains on the overall growth kinetics is primarily
dependent on phase-fraction of the microstructure. In a duplex system
characterised equal volume-fraction of phases, identical coefficients of
determination implies that both $\alpha$\- and $\gamma$-grains similarly
influence the evolution kinetics of the entire microstructure. On the other
hand, noticeable disparity between $\chi_{\alpha}$ and $\chi_{\gamma}$ is
observed in duplex microstructures with varying volume-fraction of constituent
phases. Moreover, Fig. 5 shows that this inequality in the coefficients of
determination becomes more pronounced with increase in the difference between
the volume-fraction of the phases in duplex microstructure. While the
coefficient of determination pertaining to major phase-$\gamma$ exhibits a
relatively marginal change, and continues to remain noticeably greater,
$\chi_{\alpha}$ progressively decrease with reduction in the volume-fraction
of the corresponding minor-phase grains. In other words, Fig. 5 unravels that,
in duplex systems with unequal volume-fraction of phases, the overall growth
rate of the entire microstructure ($d\bar{R}/dt$) is primarily influenced by
the evolution kinetics of the major-phase grains ($d\bar{R}_{\gamma}/dt$).
Furthermore, it is evident from the illustration that the dominance of the
major phase in effecting the overall growth kinetics becomes more definite
with increase in the corresponding volume-fraction (or decrease the amount of
minor phase).

---
Figure 6:  Progressive increase in the average radius of phase-$\alpha$ and
-$\gamma$ grains, $\bar{R}_{\alpha}^{3}$ and $\bar{R}_{\gamma}^{3}$, with time
for various duplex systems with characteristic phase-fraction.

####  3.1.4 Verifying the statistical claim

In order to substantiate the understanding rendered by the analyses based on
coefficient of determination, the temporal change in the average radius of the
phase-associated grains ($\bar{R}_{\alpha}$ or $\bar{R}_{\gamma}$) and entire
microstructure ($\bar{R}$) are studied in a conventional manner. In Fig. 6,
the progressive increase in the average radius of major- and minor-phase
grains with time, in duplex systems with varying phase-fractions, are
cumulatively presented. Since the evolution of different duplex
microstructures are considered together, for the ease of distinction, temporal
change in $\bar{R}_{\alpha}^{3}$ and $\bar{R}_{\gamma}^{3}$ is adopted for
this illustration.

Consistent with the mechanism of evolution, it is observed that the minor
phase-grains in system with the minimal volume-fraction ($\alpha 10$) grows at
a least rate. However, the growth kinetics of the these grains noticeably
increase as the corresponding phase gains more volume in the microstructure.
Accordingly, in duplex systems with unequal phase-fractions, minor-phase
grains of $\alpha 40$ microstructure exhibits highest growth-rate, followed by
$\alpha 33$ and $\alpha 16$. On the other hand, the evolution kinetics of
major-phase grains are minimal in $\alpha 40$ system, and significantly
increase in $\alpha 33$ and $\alpha 16$ as the volume fraction of the
phase-$\alpha$ reduces. Moreover, the maximum growth-rate in $\gamma-$phase
grains is observed in microstructure with minimal volume of minor phase,
$\alpha 10$. Owing to the influence of volume fraction, which governs the
kinetics through the diffusion paths, the disparity in the temporal change in
average radius of the major- and minor-phase grains becomes more evident as
the inequality in phase-fraction increases. In other words, as shown in Fig.
6, the progressive change in $\bar{R}_{\alpha}$ and $\bar{R}_{\gamma}$ with
time is notably far-apart in $\alpha 10$ system when compared to the rest.
Nevertheless, this separation gets reduced with the increase in the volume-
fraction of minor phase-$\alpha$.

---
Figure 7:  Temporal change in the average radius of entire duplex systems,
$\bar{R}^{3}$, with varying phase-fractions during grain growth.

While Fig. 6 can be appropriately discussed to affirm the consistency of the
present approach, equipped with the understanding on how evolution of
individual phase-grains relate to the growth rate of entire microstructure,
the general trend in the overall growth kinetics can be predicted from it.
Analyses based on coefficient of determination, in Fig. 5, unravels that the
growth-rate of duplex microstructure is predominantly influenced by evolution
kinetics of major-phase grains. Moreover, the effect of the minor-phase grains
decreases as their corresponding volume-fraction reduces. Adopting these
insights, and given that in Fig. 6 $\gamma$-grains of $\alpha 10$ exhibit
maximum growth rate, it can be predicted that the overall growth kinetics of
the corresponding microstructure will be noticeably greater than the other
duplex systems considered in this study. Furthermore, it can also be stated
that, since the volume fraction of minor-phase continues to be significantly
lower in $\alpha 16$ and $\alpha 33$, the overall growth will be dominated by
the $\gamma$-phase grains, and their kinetics will correspondingly follow the
$\alpha 10$ microstructure. Finally, considering that the volume of
phase-$\alpha$ is close to major-phase in $\alpha 40$, based on Fig. 5, it can
be suggested that this duplex microstructure will exhibit the least rate of
evolution.

In order to verify the accuracy of the above predictions, emerging from the
understanding of coefficient of determination, the overall growth-rate
exhibited by different duplex microstructures is cumulatively presented in
Fig. 7. It complete adherence to the prediction, it is observed that, in
duplex microstructures with unequal phase-fraction, maximum and minimum
growth-rate respectively pertains to $\alpha 10$ and $\alpha 40$
microstructure. Additionally, the kinetics of evolution exhibited by $\alpha
16$ and $\alpha 33$ lie in between the maximum and minimum, with the former
noticeably greater than the later. Ultimately, Fig. 7 affirms that, in duplex
systems characterised by unequal volume-fraction of constituent phases, the
overall grain-growth kinetics is primarily governed by the evolution-rate of
the major-phase grains. This influence of the major-phase grains gets
increasingly dominant with increase in its volume-fraction.

####  3.1.5 Phase fraction and growth rate

---
Figure 8:  Change in the kinetics coefficients that governs grain growth with
the variation in the phase-fractions of evolving duplex microstructures.

Having realised that, for a given duplex microstructure, the overall grain-
growth rate ($d\bar{R}/dt$) is predominantly dictated by the kinetics adhere-
to by the major-phase grains ($d\bar{R}_{\gamma}/dt$), attempts are made to
relate the varying phase-fraction to the observed growth-rate across different
systems. Introducing chemical-inhomogeneity amongst the grains, in a
polycrystalline system, shifts the governing mechanism, irrespective of the
volume-fraction of the phases. Kinetics rendered by this mechanism in
multiphase microstructure complies with the power-law characterised by the
exponent $n=3$. Considering that the exponent remains unaltered reflecting the
governing mechanism, despite the varying phase-fraction in multiphase systems,
the disparity in the rate of grain growth can only be understood from the
kinetic coefficient, $k$, that relates the average radius to time.

By studying the evolution of different duplex systems considered in this
study, the corresponding kinetic coefficients are ascertained. Kinetic
coefficients associated with duplex microstructures with varying phase-
fractions are illustrated in Fig. 8. Since the growth rate of the duplex
microstructure are principally dictated by the evolution of major-phase
grains, the corresponding volume-fraction is considered for this
representation. Fig. 8 indicates that with increase in the volume of a major
phase in duplex system, grain growth in the system occurs at an higher rate.
In order definitively understand the influence of the $\gamma$-fraction on the
kinetic coefficient exhibited by the corresponding duplex microstructure, non-
linear regression technique is adopted and a relation expressed as

$\displaystyle
k_{\text{dup}}=A_{\text{dup}}+B_{\text{dup}}\exp(C_{\text{dup}}\dot{V}_{\gamma}),$
(9)

is realised, where $V_{\gamma}$ is the volume fraction of the major
phase-$\gamma$. The constants $A_{\text{dup}}$, $B_{\text{dup}}$ and
$C_{\text{dup}}$, for the present consideration, respectively assume the value
of $537.3$, $3.8\times 10^{-3}$, and $0.13$. In the above relation, it is
vital to note that $k_{\text{dup}}$ indicates the kinetic coefficient adhered-
to by the entire duplex system during grain growth, not the evolving major- or
minor-phases.

###  3.2 Triplex microstructures

Triplex systems are characterised by the association of individual grains, in
the polycrystalline setup, to one of the three constituent phases.
Corresponding microstructure, in this study, comprises of phases $\alpha$,
$\beta$ and $\gamma$, with $\gamma$ largely acting as the matrix or major-
phase. In the existing works, unlike duplex systems, very few three-phase
microstructures with varying phase-fractions are analysed [32]. This limited
consideration of triplex microstructure can largely be attributed to the
computational burden associated with it. Moreover, conventionally, the grain-
growth kinetics of the triplex microstructure are discussed by focusing on the
evolution of the individual phase-grains without sufficiently relating it the
overall system. In the present study, on the other hand, grain growth in
fourteen different three-phase microstructures, with varying phase-fractions,
are examined to elucidate with statistical certainty how the evolution of the
individual phase-grains effects the growth kinetics of entire triplex
microstructure.

####  3.2.1 Grain growth kinetics

---
Figure 9:  Correlation coefficient characterising the relation between time
and average radius of individual phase-grains and entire microstructure,
raised to different power ($n=1,2,3$ and $4$), for two triplex systems,
$\alpha 39\beta 22$ and $\alpha 33\beta 33$.

Despite the difference in the number of phases, grain growth in both duplex
and triplex systems are fundamentally governed by the same mechanism.
Therefore, the grain growth in three-phase systems, which is dictated by the
diffusion of chemical components, adheres to the power law with the exponent,
$n=3$. In order to ensure that the grain growth in triplex systems is
accurately modelled by the present approach, a validation technique adopted
for duplex microstructures, is extended. Correspondingly, the temporally-
varying average radius of the individual phase-grains, $\bar{R}_{\alpha}$,
$\bar{R}_{\beta}$ and $\bar{R}_{\gamma}$, and overall microstructure,
$\bar{R}$, are raised to different powers ($n=\\{1,2,3,4\\}$) and related to
time, $t$. Correlation coefficient characterising the different relations are
ascertained for two triplex microstructures, $\alpha 33\beta 33$ and $\alpha
39\beta 22$, and are graphically illustrated in Fig. 9.

It is evident in Fig. 9 that correlation coefficient relating the average
radius to the time is highest when $n=3$ for both individual phases, and
overall microstructure. The maximum correlation exhibited by the cube of the
different average radii, $\bar{R}_{\alpha}$, $\bar{R}_{\beta}$,
$\bar{R}_{\gamma}$ and $\bar{R}$, with time implies that the growth of the
individual phase-grains, and the entire microstructure, are predominantly
governed by the long-range diffusion of the chemical components.

####  3.2.2 Ascertaining governing factor

Considering that grain growth in both duplex and triplex system are
predominantly dictated by the diffusion of the chemical components, phase-
fraction renders identical influence on the evolution of individual phases. In
other words, when certain phase(s) assumes minor volume-fraction in the three-
phase microstructure, owing to relative increase in the length, and
complexity, of the diffusion path, the growth of the corresponding grains are
stunted. On the other hand, the evolution kinetics is enhanced when the volume
of the phase(s) is dominant in the multiphase systems. Apart from these
generalised understanding, existing report rarely offer any further insights
on the grain-growth kinetics in triplex microstructures. Particularly, similar
to duplex system, sufficient consideration has not been rendered to relate the
growth kinetics of the individual phases to the evolution of the entire
triplex system. To that end, in this analysis, the impact of the growth rate
of individual phase-grains on that entire three-phase microstructure is
examined by ascertaining the corresponding coefficient of determination.

The instantaneous growth rate for constituent phase-grains,
$d\bar{R}_{\alpha}/dt$, $d\bar{R}_{\beta}/dt$ and $d\bar{R}_{\gamma}/dt$,
along with the entire triplex microstructure, $d\bar{R}/dt$, is determined by
monitoring temporal change in the respective parameter. These instantaneous
growth kinetics of the individual phase-grains are related to that of the
entire microstructure, and the corresponding coefficient of determination is
estimated through the Eqn. (8). For each system, three distinct coefficients
of determination, $\chi_{alpha}$, $\chi_{beta}$ and $\chi_{gamma}$, are
estimated, reflecting the characteristic feature of the triplex
microstructure. These coefficients of determinations are related to the phase-
fractions of the microstructure, and illustrated in Fig. 10.

---
Figure 10:  Change in coefficient of determination, typifying the influence of
growth rate of individual phase-grains on the evolution kinetics of entire
microstructure, with variation in the volume fraction of constituent phases.

In the triplex systems considered in the present study, the volume-fraction of
phase-$\gamma$ reaches as low as 33% despite being the major phase. Such
volume-fraction of phase-$\gamma$ is noticed in triplex microstructure
characterised by equifraction of phases. Furthermore, in some system like
$\alpha 45\beta 10$, the phase-$\alpha$ assumes a volume-fraction of $45\%$,
in spite of being one of the minor phases. This, and similar, understanding of
phase-fraction is vital to investigate the coefficients of determination
presented in Fig. 10.

Fig. 10 unravels that, even though the volume-fraction of the major-phase
varies noticeably across different triplex systems, the corresponding
coefficient of determination, $\chi_{gamma}$, continues to remain the high.
Given that phase-$\gamma$ stays as a major-phase, despite the change in phase-
fractions, the high values of $\chi_{gamma}$ can be attributed to the dominant
volume of the respective grains. In other words, analogous to the duplex
microstructure, the evolution kinetics of major-phase grains offer relatively
greater influence on the overall growth-rate exhibited by the entire triplex
microstructure. Furthermore, Fig. 10 suggests that the coefficient of
determination of the minor-phases, $\alpha$ and $\beta$, noticeably increases
as their corresponding volume-fraction raises. Particularly, as the volume of
phase-$\alpha$ gets as dominant as $\gamma$, in a triplex system, identical
coefficient of determination is rendered, $\chi_{alpha}=\chi_{gamma}$. On the
other hand, when the volume of the phases are minimal, the respective
coefficient of determination assumes least value. Ultimately, it is evident
from Fig. 10 that the influence of the individual phase-grains on the overall
growth kinetics depends largely on the corresponding volume-fraction. In a
triplex system with unequal volume-fraction of phases, the growth rate of the
entire microstructure, $d\bar{R}/dt$, is predominantly governed by the
evolution of the major-phase grains, $d\bar{R}_{\gamma}/dt$. When volume of
two phases are dominant in a three-phase system, the growth rate of both these
phase-grains offer identical influence on evolution of the microstructure. The
contribution of a given phase-grains to the overall evolution kinetics,
$d\bar{R}/dt$, becomes least, when its volume-fraction are minimal. Based on
the understanding rendered by Fig. 10, as demonstrated for duplex systems
(Fig. 7), the growth-kinetics of a triplex microstructure, in relation to
others with varying phase-fractions, can be predicted from the temporal change
in the average radius of the corresponding phase-associated grains.

####  3.2.3 Interdependency in the evolving phases

In duplex systems, since the grains of the polycrystalline microstructure are
associated with either of the two constituent phases, the evolution of a
particular phase-grains, and its kinetics, are inherently bound to the other.
However, the same interdependency cannot be expected in triplex systems,
wherein the grains can be associated with one of the three possible phases.
Moreover, in three-phase microstructures, level of influence offered by one
evolving phase-grains on the rest of the phase-associated grains has not been
conscientiously addressed yet. Therefore, by examining the temporal change in
the average radius of a particular phase-grains in relation to the others, the
interdependency exhibited between the phases, in triplex microstructures,
during grain growth is elucidated.

---
Figure 11:  Correlation coefficient explicating the interdependency between
the growth rate of different phase-grains, $d\bar{R}_{\alpha}/dt$,
$d\bar{R}_{\beta}/dt$ and $d\bar{R}_{\gamma}/dt$, during grain growth of four
different triplex system with varying phase-fractions.

Instead investigating all the fourteen triplex microstructure to explicate the
effect of one evolving phases on the other, the systems are categorised based
on phase-fraction, and one microstructure from each category is analysed.
Since the coefficient of determination, which quantifies the effect of an
evolving-phase grains on the growth of the overall triplex microstructure,
depends on volume fraction, the phase-fraction based grouping is deemed
reasonable. Apart from the equifraction system, $\alpha 33\beta 33$, wherein
all the constituent phases largely occupy similar volume, the remaining
systems can be categorised as ‘equi-major ’ , ‘equi-minor ’ and ‘non-
equifraction ’ triplex microstructures. While, in equi-major system, volume of
one of the minor phase is equal to that of the major-phase
($V_{\alpha}=V_{\gamma}$), the volume fraction of minor phases are identical
in equi-minor microstructures ($V_{\alpha}=V_{\beta}$). Moreover, the non-
equifraction system stand in direct contrast to the equifraction
microstructure, and characterised by totally unequal volume-fraction of the
constituent phases ($V_{\alpha}\neq V_{\beta}\neq V_{\gamma}$).

In order understand the degree of interdependency between the evolving phases,
in addition to equifraction microstructure, grain growth exhibited systems
$\alpha 36\beta 22$, $\alpha 45\beta 10$ and $\alpha 08\beta 08$ pertaining to
non-equifraction, equi-major and equi-minor, respectively, are analysed. Given
that the primary focus of the present investigation is not to quantify the
effect of an evolving-phase grains on the rest, but rather to qualitatively
realise the degree of interaction between two phases, during grain growth in a
triplex systems, coefficient of determination is not estimated. However,
alternatively, the growth rate of different phase-associated grains are
estimated, $d\bar{R}_{\alpha}/dt$, $d\bar{R}_{\beta}/dt$ and
$d\bar{R}_{\gamma}/dt$, and are related to (or plotted against) each other.
Correlation coefficient characterising the relation between the growth rate of
two phase-grains are realised for pre-determined triplex microstructures, and
graphically illustrated in Fig. 11.

Before elucidating level of interaction between two evolving phase-grain in a
given triplex microstructure, based on Fig. 11, it is exceedingly critical to
realise the variation in the range of correlation coefficient across the
different systems. Particularly, as opposed to the maximum value, which
remains constant at unity, the least value of the correlation coefficient
changes with phase-fraction. Correspondingly, Fig. 11 unravels that the lowest
correlation-coefficient in the equifraction system is maximum ($0.84$) when
compared to the rest of the triplex microstructures, and it is respectively
followed by non-equifraction ($\alpha 36\beta 22$) and equi-major ($\alpha
45\beta 10$) microstructures, with the absolute minimal exhibited by the equi-
minor system ($\alpha 08\beta 08$). This significant disparity in the lowest
value of correlation coefficient emphasis the importance of considering the
context, $i.e$ the correlation-coefficient range, while interpreting the
interaction between two-phases in a given microstructure during grain growth.
In other words, apparently least interdependency between the two-phase grains
in a equifraction system would translate to a strong interaction in the
context of equi-minor triplex microstructure.

The graphical representation of correlation coefficient, in Fig. 11, that
indicates level of influence the growth rate of one evolving phase-grain has
on the other, during grain growth in triplex microstructures, unravels few
similarities and dissimilarities across the systems with varying phase-
fractions. It is evident in this illustration that, irrespective of the nature
of the three-phase system, the correlation coefficient relating the growth
rate of $\alpha$\- and $\beta$-phase grains are minimal, within a given
microstructure. In other words, during grain growth in a triplex system, the
evolution kinetics of one minor-phase grains imposes the least effect on
growth rate of other low-volume phase-grains. Despite being equifraction,
largely owing the manner in which the triplex microstructure is initialised,
such effect is also observed in $\alpha 33\beta 33$ microstructure. However,
it is vital to note that the least correlation in equifraction system is
tantamount to noticeable interaction in relation to other triplex
microstructures. Therefore, in $\alpha 33\beta 33$ system, the evolution
kinetics of one-phase grains is generally interlinked with grains of the other
phases, but this interaction is least between phase-$\alpha$ and -$\beta$.

Within a triplex system, during grain growth, Fig. 11 suggests that minimal
interdependency following two minor-phases is observed between the minor-phase
and the matrix grains, irrespective of the phase-fractions. The only exception
is the equi-minor system wherein the volume-fraction of the minor-phases are
identical. Furthermore, in all triplex systems, the growth rate of minor-phase
grains with relatively greater volume-fraction when compared to the other
($V_{\alpha}>V_{\beta}$ ) is strongly coupled with the evolution of the major-
phase grains. Ultimately, the study of interdependency between the rate of the
evolving phase-grains during grain growth in triplex systems, using
correlation coefficients, unravel that a general trend is observed in three-
phase microstructures irrespective of the phase-fractions. If we distinguish
the constituent phases as minor- , inter- and major-phase depending on their
corresponding volume fraction, which in the present study is $\alpha$,$\beta$
and $\gamma$ respectively, then the least interaction during grain growth is
exhibited by the minor- and inter-phase grains. While the growth rate of
major-phase grains are considerably interlocked with the inter grains, the
effect of the minor phase on the matrix is comparatively lower. In other
words, in a given triplex microstructure, the level of influence offered by
the evolution rate of one phase-grains on the other, during grain growth, is
primarily dictated by the their corresponding volume fractions. When the
volume fraction of two phases are minimal, their degree of interaction is also
minimal, whereas a considerable dependency is noticed when the volume of the
two phases are dominant in view of the third.

This analysis on the interdependency of the kinetics of evolving phases in
triplex system, during grain growth, unravels that, for expressing the overall
growth rate of a three-phase microstructure, the two ideal variables, with
least multicollinearity, are the evolution rate of the grains of minor phases.

---
Figure 12:  Effect of phase-fraction on the kinetic coefficient governing the
rate of grain growth in triplex systems.

####  3.2.4 Phase fractions and growth rate

A principal insight rendered by the present study is that, in grain growth,
the evolution rate of a multiphase system is primarily governed by the volume-
fraction of its constituent phases through their respective growth kinetics.
Therefore, by relating the phase-fraction of fourteen different triplex
microstructures, considered in this work, to its corresponding growth
kinetics, an attempt to extract a generalised expression. Unlike in duplex
system, since triplex microstructures are characterised by three constituent
phases, the corresponding evolution rate is dictated by two independent
variables, $i.e$ volume fractions. By directly relating the volume fraction,
instead of the evolution rate of individual phase-grains, to the growth
kinetics exhibited by the triplex systems, the question of multicollinearity
is obviated.

Based on the proportionality exhibited by the phase-fraction and growth
kinetics in duplex microstructure, multiple non-linear regression is employed
to relate the volume-fraction of constituent phases to the corresponding grain
growth-rate exhibited by the entire three-phase microstructures. Strictly
owing to the different phase-fractions considered in the present analysis,
volume fraction phase-$\alpha$ and -$\gamma$ are considered as the independent
variables. Expression rendered by multiple non-linear regression that relates
phase-fractions to the kinetic coefficient governing the grain-growth rate of
entire three-phase microstructure reads

$\displaystyle
k_{\text{tri}}=A_{\text{tri}}+B_{\text{tri}}\exp(C_{\text{tri}}^{\alpha}V_{\alpha}+C_{\text{tri}}^{\gamma}V_{\gamma}),$
(10)

where $V_{\alpha}$ and $V_{\gamma}$ correspond to the volume fraction of
phase-$\alpha$ and -$\gamma$. Moreover, $A_{\text{tri}}$, $B_{\text{tri}}$,
$C_{\text{tri}}^{\alpha}$ and $C_{\text{tri}}^{\gamma}$ are constants whose
respective values are $268.2$, $2.2\times 10^{-7}$, $0.19$ and $0.22$. In Fig.
12 kinetic coefficient adhered to different triplex microstructures during
grain growth is plotted against the corresponding volume fraction of
phase-$\alpha$ and -$\gamma$. The curve reflecting Eqn. (10) is included in
this illustration. Noticeably good agreement between the numerical results and
the curve indicates that Eqn. (10) convincingly relates the kinetic
coefficients characterising the grain growth in isotropic triplex systems to
the phase-fractions.

##  4 Conclusion

Grain growth in polycrystalline systems can be desirous under certain
conditions, while unwelcomed in others. For instance, grain growth is induced
during processing technique to establish required average grain-size, while
noticeable measures are generally taken to avoid it during a given
application. The subjective role of grain growth extends beyond homogeneous
polycrystalline system to multiphase microstructures as well. Therefore, it
becomes vital to understand the grain-growth kinetics exhibited by highly-
applicable multiphase polycrystalline microstructures associated with duplex
and triplex systems. Particularly, generalised insights that aide in
comprehending the growth rate of multiphase microstructures with varying
phase-fraction are exceedingly critical, as they can be adopted for wide-range
of systems and application. To that end, in this study, the grain-growth
kinetics of duplex and triplex systems are studied by employing approachable
statistical techniques.

Conventionally, the grain-growth kinetics associated with multiphase systems
are discussed by considering the evolution rate of individual phase-grains and
entire-microstructure separately. Such treatments rarely offer much insights
on how the growth kinetics of individual phases relate to the overall
evolution rate exhibited by the entire system. Therefore, in the present work,
statistical tools are employed to realise the effect of growth rate of a given
phase-associated grains on the overall kinetics of evolving microstructure.
Correspondingly, it is unraveled that, during grain growth in multiphase
systems, the influence of the growth kinetics of a phase-grains on the
evolution of entire system depends on its volume fraction. In other words, in
systems with varying volume fraction of constituent phases, the evolution
kinetics of the major-phase grains predominantly govern the overall growth
rate exhibited the microstructure. Even though, when focusing on a specific
system, it might appear that the temporal change in the average radius of the
entire multiphase system lies close the evolution of the corresponding radius
of the minor-phase grains, when viewed in relation to microstructures of
varying phase-fractions, the dominance of growth kinetics of the major-phase
grains gets increasingly evident. This holistic understanding that encompasses
multiphase systems with varying phase-fractions, though arrived from
statistical treatments, in the present work, it is vindicated through
conventional representation and corresponding discussions. Moreover, in
addition to effecting the role of individual phase-grains on the evolution
kinetics of entire multiphase system, phase fraction also dictates the
interdependency between growth rates of different phase-grains. During grain
growth, evolution kinetics of grains of two phases are largely independent
when their corresponding volume-fractions in the multiphase system are
minimal. On the other hand, noticeable interaction is observed in the grain-
growth rate of two phases that dominant the multiphase microstructure.

Understanding rendered by the present investigation can be exploited for
various purposes, however, one critical utilisation would be to alter the
grain-growth rate of a given multiphase system, with a definite phase-
fraction, by appropriately, and exclusively, varying the evolution kinetics of
the major-phase grains. To that end, in the upcoming works, attempts will be
made to substantiate the approach of modifying the grain-growth rate in
multiphase systems by employing the dominant influence of the major-phase
grains.

## Appendices

### Appendix 1: Interdependency in duplex systems

---
Figure 13:  The average radius of phase-$\alpha$ grains, at a given instance,
is plotted against the corresponding radius of matrix-phase grains for duplex
microstructure with equal volume-fraction of phases. In the subplots, for the
same system, the growth rate of individual phase-grains are related, and
instantaneous average radius of phase-$\alpha$ and -$\gamma$ grains are
plotted for duplex microstructure with $10\%$ alpha.

In a polycrystalline system, irrespective of its nature, the continuum is
established by the multiple grains present it. During the grain growth,
despite the continual disappearance of the grains, the continuum is sustained
by the growth the surviving grains. This characteristic feature of the grain
growth introduces interdependency between the evolving grains.
Correspondingly, in duplex systems, wherein the grains are associated with one
of the two-constituent phases, the evolution of phase-$\alpha$ grains are
inherently linked the grains of phase-$\gamma$. Even though the interaction
between the phase-associated grains, during grain growth of a duplex system,
can be theoretically conceived, to explicate it with a statistical certainty,
the average radius of $\alpha$-grains, at a given time $t$, is plotted against
corresponding of radius of $\gamma$ grains in Fig. 13 for two-phase
microstructure with equal volume-fraction of phases, $\alpha 50$. The trend in
this illustration indicates a inherent interlocking between the evolution of
the phase-$\alpha$ and -$\gamma$ grains during the grain growth of
equifraction duplex-microstructure. In addition to the $\alpha 50$
microstructure, the average radius of constituent phase-grains, at a given
instance, are ascertained for duplex system with $10\%$ minor phase-$\alpha$.
Similar to equifraction system, these instantaneous average radii of
phase-$\alpha$ and -$\gamma$ grains are plotted against each other, and
illustrated in Fig. 13 as a subplot. Despite the change in the phase-fraction,
in $\alpha 10$microstructure as well, a definite interaction between the
radius of major- and minor-phase grains is evident.

Besides the average radius of the phase-$\alpha$ and -$\gamma$ at a given
instance, using the same approach, the relation between the kinetics of grain
growth associated with these phases can be explicated. Correspondingly, the
evolution kinetics of the $\alpha$-grains are related to that of the $\gamma$
ones, for equifraction duplex microstructure, and are included as a subplot in
Fig. 13. This illustration unravels that even though there exists a
perceivable interdependency between the instantaneous kinetics of
phase-$\alpha$ and -$\gamma$ grains, it is not as straightforward as the
average radius

### Appendix 2: Effect of individual phase-grains kinetics on growth-rate of
entire system

---
Figure 14:  The instances grain-growth rate exhibited by an individual phase-
grains are plotted with respect to that of the entire duplex microstructure
with equal volume fraction of phases. In (a) the kinetics of $\alpha$-grains
are related to the grain-growth rate of the entire microstructure, while
$\gamma$-grains evolution kinetics are considered (b). The corresponding
outcomes for $\alpha 10$ microstructure with $10\%$ of minor phase-$\alpha$
are included as subplots.

One of the primary aim of the present investigation is to realise the effect
of individual phases on grain-growth rate of entire two-phase microstructure.
Particularly, the role of evolution kinetics of a given phase-grains on the
overall growth rate of duplex system. To that end, the kinetics of evolution
exhibited by phase-$\alpha$ grains, at a given instance, is related to the
overall growth rate of equifraction duplex-system, and plotted in Fig. 14a.
The graphical illustration relating the evolution rate of phase-$\alpha$
grains and duplex microstructure with $10\%$ minor phase is included as a
subplot.

Fig. 14a unravels that the growth rate of phase-$\alpha$ grains imposes a
definite influence on the overall kinetics exhibited by the equifraction-
duplex system, $\alpha 50$, during grain growth. On the other hand, the
subplot of the corresponding illustration, which pertains to two-phase
microstructure with $10\%$ of phase-$\alpha$, indicates a relation between
$d\bar{R}_{\alpha}/dt$ and $d\bar{R}/dt$, it is not as definite as one noticed
in the equifraction microstructure. It is the degree of inter-relation between
the kinetics of individual phase-grains, and overall growth-rate of a given
duplex microstructure, which varies with the phase-fraction, is realised by
coefficient of determination. In other words, while the coefficient of
determination relating the $d\bar{R}_{\alpha}/dt$ and $d\bar{R}/dt$ will be
higher for equifraction duplex microstructure, in $\alpha 10$ system it will
assume a relatively low value reflecting a not so definite relation between
the kinetics.

In Fig. 14b the grain-growth kinetics of phase-$\gamma$ grains and entire
duplex system, with equal volume-fraction of phases, is plotted against each
other. As a subplot, the corresponding relation between $d\bar{R}_{\gamma}/dt$
and $d\bar{R}/dt$ for $\alpha 10$ duplex microstructure with $10\%$ of
phase-$\alpha$ is illustrated. Unlike the influence of phase-$\alpha$ on
overall growth-kinetics in Fig. 14a, a highly definite relation is observed
between the evolution rate of phase-$\gamma$ grains and entire microstructure
in both equifraction and $\alpha 10$ duplex microstructure. Consequently, the
corresponding values of coefficient of determination will largely be
independent of the phase-fraction.

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