# Comparison of grain growth mean-field models regarding predicted grain size
distributions

Marion Roth
Mines Paris
PSL University
Centre for Material Forming (CEMEF), UMR CNRS
06904, Sophia Antipolis, France
<EMAIL_ADDRESS>
& Baptiste Flipon
Mines Paris
PSL University
Centre for Material Forming (CEMEF), UMR CNRS
06904, Sophia Antipolis, France
<EMAIL_ADDRESS>
& Nathalie Bozzolo
Mines Paris
PSL University
Centre for Material Forming (CEMEF), UMR CNRS
06904, Sophia Antipolis, France
<EMAIL_ADDRESS>
& Marc Bernacki
Mines Paris
PSL University
Centre for Material Forming (CEMEF), UMR CNRS
06904, Sophia Antipolis, France
<EMAIL_ADDRESS>
Corresponding author.

(September 15, 2023)

###### Abstract

Mean-field models have the ability to predict grain size distribution
evolution occurring through thermomechanical solicitations. This article
focuses on a comparison of mean-field models under grain growth conditions.
Different microstructure representations are considered and discussed,
especially regarding the consideration of topology in the neighborhood
construction. Experimental data obtained with a heat treatment campaign on a
316L austenitic stainless steel are used for material parameters
identification and as a reference for model comparisons. Mean-field models are
also confronted to both mono- and bimodal initial grain size distributions to
investigate the interest of introducing neighborhood topology in
microstructure predictions models. This article exposes that improvements in
the predictions are obtained in monomodal cases for topological models. In
bimodal test, no comparison with experimental data were performed as no data
were available. But relative comparisons between models indicate few
differences in predictions. The interest of neighborhood topology in grain
growth mean-field models gives overall small improvements compared to
classical mean-field models when comparing implementation complexity.

_Keywords_ Mean-field model $\cdot$ Grain growth $\cdot$ Grain size
distribution $\cdot$ Topology $\cdot$ Neighborhood description

##  1 Introduction

The phenomenon of grain growth takes place in metallic materials when their
are submitted to a heat treatment. Considered as the only phenomenon
occurring, materials are assumed free of any stored energy (i.e. low
dislocation density) and the driving pressure for grain boundary (GB)
migration arises from the minimization of the GB surface energy leading to a
curvature flow problem. At the polycrystalline scale, the GB motion is
generally described by $v=M_{GB}\left|P\right|$ with $v$ the velocity norm of
the boundary [1], $M_{GB}$ its mobility and $P=-\gamma_{GB}\kappa$ with
$\gamma_{GB}$ the GB energy and $\kappa$ the trace of the curvature tensor.

Since 70 years, models have largely been developed in order to predict
microstructure changes under thermomechanical treatments and their impact on
macroscopic properties. At the polycrystalline scale, three types of models
can be found in the literature: phenomenological, mean-field, and full-field
models. Phenomenological approaches are classically based on an experimental
database and correspond to the extraction of a mathematical trend from what is
observed experimentally. Such models are restricted to the set of
thermomechanical conditions and to the material experimentally investigated
[2, 3]. The last two model types are based on a more generic approach, since
they use physical equations to predict microstructure evolution. Full-field
models, at the mesoscopic scale, give access to a complete description of the
system where each individual grain and its topology are taken into account [4,
5, 6]. It has the advantage to be able to deal with local heterogeneities.
However, their major drawback is to be very costly in terms of computing time.

Mean-field models, on the other hand, keeps this physically based
implementation with a similar set of equations, but the general description of
the microstructure is simplified [7, 8, 9, 10, 11, 12, 13]. They also present
competitive computation time when compared to full-field models. The original
definition of a mean-field model considers the evolution of a microstructure
described by mean quantities. The work of Burke and Turnbull [14] provides the
simplest version of such models. In this case, the microstructure is solely
defined by its mean grain radius $\bar{R}$. The mean grain size (MGS) quantity
is commonly determined by the (arithmetic) mean of the equivalent diameter
($\overline{ED}=2\bar{R}$) where $ED_{i}$ is defined in 2D, resp. in 3D, as
the diameter of a circle, resp. a sphere, having the same area ($A_{i}$),
resp. volume ($V_{i}$), as the considered grain $G_{i}$:

$\bar{R}=\frac{\overline{ED}}{2}=\frac{1}{2N}\sum_{i=1}^{N}ED_{i},\text{ i.e
}\bar{R}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{A_{i}}{\pi}\right)^{1/2}\text{
in 2D, and
}\bar{R}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{3V_{i}}{4\pi}\right)^{1/3}\text{
in 3D,}$ (1)

with $N$ the number of grains in the microstructure.

In Burke and Turnbull (B&T) context, the GB velocity norm is considered
proportional to the inverse of $\bar{R}$. Hillert model [7] introduces the
consideration of the grain size distribution (GSD) in mean-field models. The
microstructure is sampled in a representative distribution, where each bin is
assigned to a given ED value and a frequency. In the GSD, a bin is commonly
called grain class, each of them being virtually composed of many grains
indicated by the frequency. For every class, one can define a representative
grain having the characteristics of the associated grain class. Hillert
formalism represents the microstructure as a grain embedded in a Homogenous
Equivalent Medium (HEM) that is characterized by the MGS of the distribution.
The size evolution of a considered class is deduced from the difference in
curvature radius between the current class and that of the HEM. Later on,
statistical models, as the one developed by Abbruzzese et al. [8, 15] modify
the definition of the HEM. Rather than being only associated with a mean grain
radius, the HEM is replaced by contact surfaces defined for all grain classes
of the microstructure. Each grain class is surrounded by a statistical medium
(SM) composed of all the grain classes of the microstructure. The contact
surface between a grain class and its surrounding neighbors is defined by a
perimeter intersection probability. One of the latest topological mean-field
models, developed by Maire et al. [13] combines several formalisms in its
neighborhood construction. This hybrid model uses both the statistical
approach initiated by Abbruzzese et al. and a deterministic number of
neighbors ruled by the imposed bijectivity of neighborhood assignment. To the
authors knowledge, the interest of these different views concerning the
neighborhood description was never discussed in the state of the art for GG
modeling. The main purpose of this work is to compare mean-field models of
different microstructure descriptions to determine if semi-topological
approaches are of interest to better describe GSD evolution in the context of
grain growth.

This article is constructed as follows: first, mean-field GG models are
introduced. A detailed description of the microstructure formalism in Hillert,
Abbruzzese et al. and Maire et al. models is made. Then optimized data
parameters for the use of these models are determined and discussed. The last
section is dedicated to a distribution comparison of the Maire et al. model to
the mean-field models of Hillert, Abbruzzese et al. and to experimental data.

##  2 Mean-field models

This section recalls the main equations and details the microstructure
description of the previously introduced models for the GG phenomenon.

###  2.1 Burke and Turnbull model

B&T model [14] illustrates the original meaning of mean-field models as the
microstructure description is reduced to mean geometric considerations.
Indeed, the rate of GG ($d\bar{R}/dt$) is assumed proportional to $1/\bar{R}$
[1] such as:

$d\bar{R}=M_{GB}P_{c}dt\text{, with
}P_{c}=\frac{\alpha\left(d-1\right)\gamma_{GB}}{\bar{R}},$ (2)

where $\alpha$ is a constant, $\gamma_{GB}$ is the GB energy, $M_{GB}$ is the
GB mobility, and $d$ the space dimension. In the B&T analysis, $\gamma_{GB}$
and $M_{GB}$ are assumed to be constant in time and space.

A parabolic expression can then be derived from eq. (2) and represents the
time dependence of $\bar{R}^{2}-{\bar{R}_{0}}^{2}$:

$\bar{R}^{2}-{\bar{R}_{0}}^{2}=2\left(d-1\right)\alpha M_{GB}\gamma_{GB}t,$
(3)

where $\bar{R}_{0}$ is the initial mean grain radius. An extension of this law
is classically preferred in the literature :

$\bar{R}^{2}-{\bar{R}_{0}}^{2}=\tilde{\alpha}M_{GB}\gamma_{GB}t^{n},$ (4)

with $n$ and $\tilde{\alpha}$ two constants to be fitted thanks to
experimental data.

###  2.2 Hillert model

The GG mean-field model proposed by Hillert [7] relies on considering each
grain class inside a common HEM. The microstructure description is illustrated
in fig. 1 where the representative grain of a given grain class $R_{i}$ is
surrounded by a HEM defined by the mean grain radius of the microstructure
$\bar{R}$. Compared to B&T model, the driving pressure term is modified
following a more local approach. $P_{c}$, for the class $i$, is defined as a
function of the curvature radius of the grain class $i$:

$dR_{i}=\frac{(d-1)}{2}M_{GB}\gamma_{GB}\left(\frac{1}{\bar{R}}-\frac{1}{R_{i}}\right)dt.$
(5)

Thus, the grain $i$ will shrink when $\bar{R}>R_{i}$, will grow if
$\bar{R}<R_{i}$ and will be stable when $\bar{R}=R_{i}$.

Figure 1:  Schematic representation of the microstructure in the Hillert
model.

###  2.3 Abbruzzese et al. model

In the statistical model proposed by Abbruzzese et al. [8, 15] in 1992, the
authors suggested an evolution of the Hillert model by introducing a
statistically constructed medium (SM) composed of all grain classes of the
microstructure as illustrated in fig. 2(a). The contribution of all neighbors
being weighted by a statistical coefficient: the contact probability.

In 2D, considering the grain class $i$, the contact between $i$ and its
neighbor grain classes is defined by a fraction of the neighbor class
perimeter. The probability of a grain class $j$ to belong to the SM is only
depending on its own size (grain radius – $R_{j}$) and will be the same for
all grain class $i$ considered. This represents what would be the chances or
probabilities that a grain of the $j\textsuperscript{th}$ class in the
microstructure intersects the representative grain $i$ as schematized in fig.
2(b). The contact probability $p_{j}$ is then given by a ratio between the
$j\textsuperscript{th}$ grain perimeter and the sum of all the grain
perimeters of the microstructure. Along with the grain size, the number
frequency of each grain class is also taken into account. $p_{j}$ expression
is written as follows:

$p_{j}=\frac{N_{j}R_{j}}{\sum_{k=1}^{n}{N_{k}R_{k}}},$ (6)

with $n$ the total number of grain classes in the microstructure and, $\forall
k\in\llbracket 1,n\rrbracket$, $N_{k}$ the number of grains belonging to the
grain class $k$.

(a)

(b)

Figure 2: Statistical neighborhood construction of the 2D GG model of
Abbruzzese et al.: (a) description of the statistical medium, and (b)
illustration of the contact probability $p_{j}$ concept.

With such a kind of explicit description of the surrounding, the GB migration
can be achieved locally between the class $i$ and each of its neighbors. In
the context of 2D-GG ($d=2$), the difference of driving pressure between a
grain class $i$ and its neighbor $j$ can be derived from the Hillert equation
(eq. (5)) and is then given by:

$dR_{(i,j)}=\frac{1}{2}M_{GB}\gamma_{GB}\left(\frac{1}{R_{j}}-\frac{1}{R_{i}}\right)dt.$
(7)

The global variation of grain size undergone by the grain class $i$ is the sum
of the local variations with every $j\textsuperscript{th}$ neighbor. The local
GB migration can be generalized to all the $j\textsuperscript{th}$ neighbors
of the grain class $i$ using the contact probability $p_{j}$. The total rate
of evolution with respect to time for the radius of the representative grain
$i$ can then be written as:

$dR_{i}=\sum_{j=1}^{n}p_{j}dR_{(i,j)}=\frac{1}{2}M_{GB}\gamma_{GB}\left(\sum_{j=1}^{n}\frac{p_{j}}{R_{j}}-\frac{1}{R_{i}}\right)dt.$
(8)

###  2.4 Maire et al. model

Maire et al. [13] proposed a 3D mean-field model to simulate microstructure
evolution under thermomechanical solicitations. Physical mechanisms such as
GG, discontinuous dynamic recrystallization and post-dynamic recrystallization
can be modeled. It is based on previous works of Bernard et al. [11] and
Beltran et al. [16] who developed a mean-field model with two HEMs, one medium
associated to non recrystallized grains and the other to recrystallized ones,
each of these media being characterized by their MGS. Maire et al. work
focused on the introduction of neighborhood topology in the Bernard-Beltran
previously defined formalism. A specific neighborhood is proposed for each
grain class of the microstructure. Each class $i$ is characterized, in the
context of GG, by two main properties: a grain radius $R_{i}$ and a number of
grains belonging to this class $N_{i}$. Microstructure and its evolution are
described as detailed below. The following equation links the radius variation
$dR_{(i,j)}$ between two representative grains to the volume variation
$dV_{(i,j)}$ associated to the GB migration:

$dV_{(i,j)}=dR_{(i,j)}\times S_{c(i,j)}=dR_{(i,j)}\times p_{(i,j)}\times
S_{R_{i}},$ (9) $\text{with
}dR_{(i,j)}=M_{GB}\gamma_{GB}\left(\frac{1}{R_{j}}-\frac{1}{R_{i}}\right)dt\text{,
as }d=3\text{ (see eq. \eqref{eq:Hillert})},$ (10)

and, with $S_{c(i,j)}$ the contact surface between class $i$ and $j$ which is
the product of $p_{(i,j)}$ the contact probability and $S_{R_{i}}$, the
remaining surface of the grain class i:

$S_{c(i,j)}=p_{(i,j)}\times S_{R_{i}}.$ (11)

Figure 3: Microstructure description in the Maire et al. model.

Fig. 3 illustrates the microstructure description considered in the Maire et
al. model.

This model considers a specific neighborhood (SN) that is built for each grain
class based on two main criteria. First, bijectivity is ensured between two
neighbor classes. For two considered neighbor grain classes $i$ and $j$, the
contact probability of grain class $i$ with the grain class $j$ is computed as
$p_{(i,j)}$ and it is automatically associated with the grain class $j$ as the
contact probability $p_{(j,i)}=p_{(i,j)}$. The second criterion imposed by
this model, intimately linked to the first one, is the volume conservation
that will be defined in more details below.

At each time increment of a grain growth simulation, the surface of the
spherical representative grain of a class is filled with a neighborhood.
Initially, each grain class has a remaining (available) surface equal to the
surface of a sphere of radius $R_{i}(t=0)$: $S_{R_{i}}=4\pi R_{i}^{2}$. During
the SN construction, the remaining available surface will decrease as
neighbors are assigned until the SN of the considered grain class is
completely filled (i.e. no more available surface).

The filling of the SN is arbitrarily selected to be done in the ascending
order of the GSD. By considering $i$ as the current class, a first neighbor
class $i+1$ is attributed to the class $i$. The corresponding contact surface
$S_{c{(i,i+1)}}$ is applied between the two considered grain classes and
$S_{R_{i}}$ is updated as follows:

$S_{R_{i}}=4\pi R_{i}^{2}-S_{c{(i,i+1)}},$ (12)

$S_{c{(i,i+1)}}$ computation will be described with more detailed below.

Figure 4: Specific Neighborhood construction in the Maire et al. model for
class $i$.

The next neighbors, $j=i+2,\ i+3,\ldots$ for class $i$, are then selected
following the ascending order of the initial GSD and a contact surface is one
more time computed. This goes with all the microstructure grain classes with
incomplete neighborhood. Once the $i\textsuperscript{th}$ grain class
neighborhood is filled. The following class, denoted $i+1$ here, becomes the
current class and so on until the whole grain distribution is scanned. As the
current class evolving in the GSD, a part of its neighborhood is already
filled up by bijectivity with classes of smaller radius. In fig. 4, the
current class $i$ considered is taken in the middle of the GSD. Green classes
represent the ones attributed to the grain class $i$ by bijectivity when
$i-1,i-2,\ldots$ were current grain classes. The blue ones are all the
remaining classes of the distribution with no complete neighborhood. These
classes are added to grain $i$ neighborhood when it is the current class.
Finally, red grain classes do not belong to the grain class i neighborhood as
their neighborhood are already complete entirely by bijectivity with smaller
radius grain classes.

As mentioned in the introduction, a bijective criterion is applied here. The
remaining surface of the grain class $i+1$, $S_{R_{i+1}}$, has already been
updated when $i+1$ has been attributed as the neighbor to the grain class
$k\text{, with }k\leq i$. That way, as the current class goes up in the
distribution a part of the neighborhood has already been filled by bijectivity
with grains having a radius smaller than the current class. After update, the
remaining surface of grain class $j$ progressively decreases. A test is
realized to check if $S_{R_{j}}>0$: if true, grain class $j$ can take new
neighbors; and if not, grain class $j$ neighborhood is complete. Class $j$ is
therefore not available anymore as a neighbor and will not take part in the
construction of the following grain classes (class $j$ will be seen as the red
grain classes from fig. 4 for all other classes)

Of course, it is important here to highlight that the order used to build the
neighborhood of each class can have an impact on the global answer of the GG
model as the neighborhood determination will be different for example if the
distribution is treated by a decreasing GSD order. This effect will be
discussed below.

The eq. (9) introduces the contact probability $p_{(i,j)}$ between two grain
classes of the microstructure. The idea of Abbruzzese et al. contact
probability has been conserved in the equation form, however the spatial
dimension of contact used is elevated to a volumic consideration as described
in the following equation:

$p_{(i,j)}=\frac{N_{j}R_{j}^{3}}{\sum_{k=1}^{n_{i}}{N_{k}R_{k}^{3}}},$ (13)

with $n_{i}$ the number of grain classes with an incomplete neighborhood when
the $i\textsuperscript{th}$ grain class neighborhood is constructed.

As presented previously, this model is described in 3D, so quantities
exchanged during GB migration are volumic. The volume variation seen by a
grain is the sum of all signed volume variations with respect to its neighbors
given by eq. (9) such as:

$\Delta V_{i}=\sum_{j=1}^{\eta_{i}}dV_{(i,j)}$ (14)

with $\eta_{i}$ the number of neighbors of the grain class $i$. As
$dV_{(j,i)}=-dV_{(i,j)}$ owing to the imposed bijectivity, the volume
conservation of the global system is ensured.

##  3 Input data for mean-field modeling

###  3.1 Material-dependent model parameters acquisition

Experimental data and material parameters identification are necessary to
calibrate a mean-field model for a given material and a given temperature
range. This section will detail the material and experimental data used for
model parameters identification.

####  3.1.1 Experimental data

A single-phase austenitic stainless steel (316L) was selected for this study.
The identification procedure of the reduced mobility ($M_{GB}\gamma_{GB}$
product) presented in this work relies on a minimal campaign of nine thermal
treatments [17, 18]. Fifteen are provided here and their conditions are
detailed in table 1. Longer annealing times from 3 to 5 hours have been
realized to validate the identification procedure at long durations. These
heat treatments have been performed using a Carbolite furnace. A thermocouple
was placed in the furnace near to the samples to control and record the
temperature evolution.

$1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$ | $1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$ | $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$
---|---|---
$30\text{\,}\min$ | $30\text{\,}\min$ | $30\text{\,}\min$
$1\text{\,}\mathrm{h}$ | $1\text{\,}\mathrm{h}$ | $1\text{\,}\mathrm{h}$
$2\text{\,}\mathrm{h}$ | $2\text{\,}\mathrm{h}$ | $2\text{\,}\mathrm{h}$
$3\text{\,}\mathrm{h}$ | $3\text{\,}\mathrm{h}$ | $3\text{\,}\mathrm{h}$
$5\text{\,}\mathrm{h}$ | $5\text{\,}\mathrm{h}$ | $5\text{\,}\mathrm{h}$
Table 1: Heat treatments campaign conditions, with orange contoured conditions
used as qualibration values, red conditions are used for both calibration and
validation and green ones are only used for validation.

These samples have been prepared for electron back scattered diffraction
(EBSD) analyses by cutting and selecting a centered observation area to avoid
any effects of surface oxides on analyses. Classical first steps of polishing
for stainless steel were realized using abrasive SiC papers, followed by
polishing with a $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diamond suspension
and final electropolishing for 25s at 10V with a solution of 10$\%$ perchloric
acid in ethanol. EBSD analyses were performed with a Carl Zeiss Supra 40 field
emission gun scanning electron microscope (FEGSEM) coupled with a Bruker
Quantax EBSD detector and the Esprit 2.3 software. A voltage of 20kV and a
$120\text{\,}\mathrm{\SIUnitSymbolMicro m}$ aperture were used. The post-
processing of EBSD data was achieved with the MTEX Matlab toolbox [19]. The
step and the cartography size have been targeted to get a representative
number of grains in the observation area. Table 2 gathers the latter described
parameters as well as the number of grains observed in each cartography. This
number of grains is calculated without taking into account twin boundaries and
using a misorientation of 15°to define a grain boundary. The minimal size to
consider an entity as a grain is superior to 5 pixels. The process of entities
removal under the set threshold and a grain boundaries smoothing were applied
considering only indexed pixels. These data give also access to 2D GSD that
will be transposed in 3D with the Saltykov algorithm (detailed in the
following section) to be used as initial input distribution in the model but
also as experimental data to compare with simulation results in section 4.
Fig. 5 displays the IPF Z maps of some of the post-treated experimental
results as well as the related histograms in number frequency.

| $T=$1000°C | $T=$1050°C | $T=$1100°C
---|---|---|---
$t$ | $h$ (µm) | $L_{x}\times L_{y}$ ($mm\times mm$) | #G | $h$ (µm) | $L_{x}\times L_{y}$ ($mm\times mm$) | #G | $h$ (µm) | $L_{x}\times L_{y}$ ($mm\times mm$) | #G
Initial | 1.49 | 1.1$\times$0.85 | 980 | 1.49 | 1.1$\times$0.85 | 980 | 1.49 | 1.1$\times$0.85 | 980
30min | 2.5 | 2$\times$1.4 | 2654 | 1.13 | 1$\times$0.7 | 534 | 3.3 | 3.7$\times$2.8 | 3509
1h | 2.5 | 2$\times$1.4 | 2078 | 3 | 3$\times$2.2 | 1964 | 3.3 | 3.7$\times$2.8 | 3590
2h | 1.13 | 1$\times$0.7 | 456 | 3 | 3$\times$2.2 | 1154 | 3.77 | 3.7$\times$2.8 | 2208
3h | 1.13 | 1$\times$0.7 | 468 | 1.13 | 1$\times$0.7 | 300 | 3.77 | 3.7$\times$2.8 | 2263
5h | 1.13 | 1$\times$0.7 | 243 | 1.13 | 1$\times$0.7 | 133 | 3.77 | 3.7$\times$2.8 | 2304
Table 2: Table gathering, for the different isothermal treatment temperatures
and for the different holding times, the post-processing details composed of
the EBSD step size ($h$), the dimensions in each direction ($L_{x}\times
L_{y}$) of analyzed areas of the sample, and the number of grains represented
in the EBSD images considering all the grains without taking into account the
twins boundaries (#G).

(a) Initial microstructure.

(b) After a heat treatment of $2\text{\,}\mathrm{h}$ at
$1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

(c) After a heat treatment of $2\text{\,}\mathrm{h}$ at
$1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

(d) After a heat treatment of $2\text{\,}\mathrm{h}$ at
$1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

Figure 5: In background: EBSD IPF Z maps of 316L microstructures of (a) the
as-received material, afer (b) 2h at 1000°C, (c) 2h at 1050°C and (d) 2h at
1100°C. Black lines denote all grain boundaries. In foreground for each image:
the corresponding 3D-GSD after an inverse Saltykov transformation.

###  3.2 Use of Saltykov algorithm to obtain a 3D GSD

Experimentally acquired EBSD data correspond to 2D slices of 3D polycrystals.
In order to be consistent with mean-field simulations results and to enable
comparisons, a 2D to 3D conversion is performed. To this end, the Saltykov
method [20] is applied to experimental GSDs. Initially, the method had been
developed to exhibit 2D section data from 3D granulometry data of spherical
particles. The inverse Saltykov method gives the possibility to transform GSD
data from a 2D histogram distribution to a 3D discrete distribution [20].
Several assumptions are present in this method which are compatible with the
topology of the considered initial 316L microstructure. The Saltykov method
has indeed been proven to be efficient on a similar equiaxed polycrystal [21].
No assumption is required regarding the shape of the input distribution, so
multi-modal distributions can also be submitted to such a method [22]. The
methodology ensures that the average of an infinite number of 2D-cuts of a
polycrystal respecting the obtained 3D-discrete distribution will converge
towards the imposed 2D-histogram distribution. However, the quality of the
methodology is, of course, also linked to the statistical representativity of
the input 2D GSD. This procedure is illustrated in fig. 5 to exhibit the 3D
distribution evolution after the 2h thermal treatments at the different
temperatures summarized in table 2. Moreover, fig. 6 illustrates for one
particular microstructure ($T=$$1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$
for $t=$$5\text{\,}\mathrm{h}$) the comparison between the 2D-histogram
distribution and the 3D obtained discrete distribution thanks to the inverse
Saltykov transformation.

Figure 6: Inverse Saltykov method illustrated on the 2D GSD of a sample heat
treated at $1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for
$5\text{\,}\mathrm{h}$, the blue histogram corresponds to the 2D GSD and the
orange discrete distribution corresponds to the obtained 3D results after the
inverse Saltykov algorithm.

####  3.2.1 GB mobility parameter identification

##### A first approximation by the use of classical B&T law

The first step of the GB mobility identification procedure is to find an
initial approximation for the reduced mobility $(M_{GB}\gamma_{GB})$, in order
to run a first mean-field computation. To this end, the historical form of the
Burke and Turnbull law [14, 1] (eq. (4) with $\tilde{\alpha}=1/2$ and $n=1$)
is used to obtain a first approximated value of the reduced mobility for each
considered temperature (see table 1).

For each heat treatment temperature, the B&T law is plotted in order to obtain
a linear dependence between $\bar{R}^{2}-{\bar{R}_{0}}^{2}$ and the time $t$.
Fig. 7 illustrates the methodology on sets of experimental points for the
three studied temperatures where the best linear fit directly provides a first
rough value for $(M_{GB}\gamma_{GB})$ called $(M_{GB}\gamma_{GB})_{ini}$.

(a)

(b)

(c)

Figure 7:  Use of the Burke & Turnbull law to obtain the first value of
$(M_{GB}\gamma_{GB})_{ini}$ for 316L at (a)
$1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$,(b)
$1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and (c)
$1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

##### Refined identification

The values $(M_{GB}\gamma_{GB})_{ini}$ are then used in Hillert and Maire et
al. models for comparison with experimental points (solid blue and red lines
respectively in fig. 8). These first simulation results are then used to
perform an optimization of the reduced mobility by calculating the $L^{2}$
error on several points between the simulation and experimental results. A
translation coefficient $c_{fit}$ is defined by a least square method to shift
the simulated curve in order to improve the correlation with experimental data
so that:

$\left(M_{GB}\gamma_{GB}\right)_{fit}=c_{fit}\times\left(M_{GB}\gamma_{GB}\right)_{ini}.$
(15)

The cost function of the least square method is as followed:

$F(c)=\sum_{i=1}^{n}f_{i}^{2}(c_{i}),$ (16)

where $n$ is the number of experimental points,
$c_{i}=\frac{t_{sim_{i}}}{t_{exp_{i}}}$ the translation coefficient between
the interpolated curve of the simulation data and the experimental points and
$f_{i}(c_{i})$ compute the $L^{2}$ errors between the simulated and
experimental point for each value of $c_{i}$:

$f_{i}(c_{i})=L^{2}_{i}(c_{i})=100\times\sqrt{\frac{\sum_{k=1}^{n}(\frac{t_{sim_{k}}}{c_{i}}-t_{exp_{k}})^{2}}{\sum_{k=1}^{n}t_{exp_{k}}^{2}}}\text{
with }\forall i\in\llbracket 1,n\rrbracket.$ (17)

Figure 8:  Curve fitting of simulation points (obtained with
$\left(M_{GB}\gamma_{GB}\right)_{ini}$) with respect to experimental points by
minimizing L2 error for Maire et al. and Hillert models at
$1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 316L.

##### Reduced Mobility data

Temperature | $1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$ | $1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$ | $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$
---|---|---|---
$M_{GB}\gamma_{GB}$($\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$) | 2.30e-15 | 1.08e-13 | 1.10e-13
Table 3: Table of the reduced mobility values $(M_{GB}\gamma_{GB})_{ini}$ for
Maire et al. model considering an ascending sorting order for the initial GSD
for the investigated temperatures for 316L.

##### Model-dependence of Reduced Mobility

Since the intrinsic GB mobility of a material is hard to quantify
experimentally, a common approach is to model the GB migration evolution (for
instance using the $v=M_{GB}P$ equation) and consider the mobility $M_{GB}$ as
a material-dependent model parameter. Depending on the definition used in the
pressure term $P$, mobility will also be a model-dependent parameter.

Fig. 8 solid lines illustrate the difference in response from Hillert and
Maire et al. models to an identical mobility value. When the mobility is
identified for each model, the respectively colored dashed lines are obtained
for both models in the same figure. Reduced mobility values at the temperature
of $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ are gathered in table 4. The
GB migration equation in these models (cf. eq. (5), (7), (10)) is rather
similar which explains that the identified reduced mobility are of the same
order of magnitude.

Model | Hillert | Abbruzzese | Maire
---|---|---|---
$M_{GB}\gamma_{GB}$($\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$) | 1.08e-13 | 1.27e-13 | 1.10e-13
Table 4: Identified reduced mobility values for the different models at
$1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 316L.

##  4 Results and discussion

This section is dedicated to numerical parameters optimization and to the
comparisons between results obtained using the different introduced mean-field
models for 316L austenitic stainless steel in GG context. Different heat
treatment conditions (temperature, duration) of table 1 are simulated and
compared with experimental GSDs.

###  4.1 Numerical parameters

As described in section 2, the three mean-field models (Hillert, Abbruzzese et
al. and Maire et al.) are based on specific media or neighborhood. These
latter may have an influence on several modeling parameters which requires a
thorough study to either optimize their value or evaluate their impact on
model predictions.

####  4.1.1 Convergence study concerning the number of grain classes
introduced in the model

An important common parameter to these models is the initial number of grain
classes introduced at the start of a simulation. Statistical representativity
is represented by two criteria: the minimum number of grain classes necessary
for a GSD to be representative of an experimental microstructure and the
representation necessity of the different grain populations existing in the
material (detection of mono- or multimodal distributions). The convergence
study performed here will focus on the first described criterion. In GG, the
number of grain classes drops in time due to capillarity effects, the
representativity of the microstructure is therefore impacted. Maire et al.
neighborhood construction relies on a good statistical representativity, as it
widens the choice of classes in the neighbors selection. A convergence study
is performed on this parameter in order to determine the minimal initial
number of grain classes necessary to obtain a reproducible final GSD. A
thermal treatment of one hour at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$
is used for this discussion and GSD results will be compared to a reference
simulation for each model. More precisely, seven simulations from 25 to 2000
initial grain classes have been run and compared to a reference simulation
where 5000 grain classes were considered. This reference is considered as
representative of experimental values. Indeed, the number of grains in the
sample area examined by EBSD analysis varies from 3500 to 130 grains with an
average around 1400 grains for all studied conditions according to table 2.
The use of 5000 initial grains classes perfectly covers the 980 grains
observed experimentally. Also, a final number of 1300 classes is obtained with
this simulation after 5h.

To be able to define at which value convergence is reached, eq. (18) gives the
relative error, at time $t$ between the reference case and the tested GSDs:

$L^{2}(t)=100\times\sqrt{\frac{\sum_{i=1}^{n_{\text{bins}}}{(S_{i}-S_{i}^{\prime})}^{2}}{\sum_{i=1}^{n_{\text{bins}}}{(S_{i}^{\prime})}^{2}}},$
(18)

with $n_{\text{bins}}$ the number of bins in the histograms used for
comparison. This number $n_{\text{bins}}$ is introduced to simplify the visual
representation of histograms and have a meaningful comparison between
simulations, a reduced number of histogram bins is then selected comparatively
to the total number of grain classes. Typically, for the following histogram
representations presented in this paper at exception of fig. 9(a),
$n_{\text{bins}}$ is set to 25. The corresponding bin width is computed from
this number. This allows to recreate a unique ECD vector of 25 visual grain
classes equally distanced from each other to rightfully compare GSDs.

In fig. 9(a), to introduce the $L^{2}$ comparison, a number of 9 bins is
selected to simplify the visual explanation. In this figure $S_{i}$ (resp.
$S_{i}^{\prime}$) corresponds, for the Maire et al. model, to the
$i\textsuperscript{th}$ bin grain class area of the GSD at
$t=$1\text{\,}\mathrm{h}$$ (resp. of the model reference GSD).

(a)

(b)

(c)

Figure 9: (a) Histogram description of the $L^{2}(1h)$ error method to analyse
convergence of simulations at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
(b) convergence study of the initial grain classes number by the computing
$L^{2}(1h)$ error at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for Hillert,
Abbruzzese et al. and Maire et al. models on 316L and (c) same convergence
study computing the $L^{2}(1h)$ error for the three models using the final
number of grain classes as x-axis.

Fig. 9(b) plots the $L^{2}($1\text{\,}\mathrm{h}$)$ evolution for the
different initial number of grain classes in each model at
$1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$. From 25 to 500 initial grain
classes $L^{2}($1\text{\,}\mathrm{h}$)$ error decreases drastically from above
100$\%$ to 3$\%$. 25 initial grain classes give rise to a
$L^{2}($1\text{\,}\mathrm{h}$)$ error superior to 100 $\%$ for all models,
traducing therefore a degradation of the statistical representativity. A
convergence threshold is set at 5$\%$, considering that convergence is reached
below that error. From 500 to 2000 initial grain classes, the simulations are
therefore converging considering the threshold. For comparisons performed in
section 4, an initial number of 1000 grains classes is chosen in order to
assure consistency and computational time efficiency. This value is also in
perfect accordance with the experimental initial number of grains of 980 as
exposed in table 2. The statistical representativity depends also on a second
parameter which corresponds to the number of grain classes present at the end
of a numerical computation.

Fig. 9(c) illustrates the same convergence study as in fig. 9(b) but where the
x-axis represents the number of final grain classes at the end of each
simulation. This shows that a minimum number of 200 remaining grain classes is
necessary to stay below a threshold of 5% of $L^{2}(1h)$ error. As the model
is working in grain classes, it can be considered that 200 classes can
describe with good accordance the range of experimental number of grains from
130 to 3500 observed in EBSD maps for the different conditions. This figure
also shows that for an identical initial number of grain classes Hillert model
conserves a higher number of remaining grain classes and therefore a better
statistical representativity.

For forthcoming GSD comparisons, the statistical representativity for heat
treatment from 2 to $5\text{\,}\mathrm{h}$ at
$1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with an initial number of 1000
grain classes was also verified thanks to the Maire et al. model as
illustrated in fig. 9(c) with the colored triangle icons. Indeed, one can see
for the three thermal treatments that the threshold for representativity
previously defined is well respected as the $L^{2}(t)$ error (comparatively to
the reference case at 5000 grains classes) remains below 5% and that the
remaining number of classes after annealing time is above 200.

####  4.1.2 Different spatial dimensions considered to define the contact
probability

##### Description of the spatial dimensions

In the original work of Abbruzzese et al. [8], the 2D contact probability is
computed based on the perimeter of the neighbor class (cf. section 2.3). This
formalism was extended to 3D by considering sphere surfaces instead of disk
perimeters in Di Schino et al. [23]. The contact probability construction
suggested by Maire et al. is a generalization of the latter formalism. This
section will focus on determining if the contact probability has a positive
effect on refining the description of Maire et al. model simulated GSDs.
Inspired from Abbruzzesse et al. work with eq. (6), four types of contact
probability can be derived from a probability in number to a volumic
probability:

$p_{(i,j)}^{m}=\frac{N_{j}R_{j}^{m}}{\sum_{k=1}^{n_{i}}{N_{k}R_{k}^{m}}}\quad\text{with
}m\in\llbracket 0,3\rrbracket,$ (19)

where $N_{j}$ is the number of grains belonging to the grain class $j$ and
$n_{i}$ the number of grain classes with an incomplete neighborhood when
$p_{(i,j)}^{m}$ is computed for the grain class $i$.

Figure 10: Description of the different contact probabilities $p_{(i,j)}^{m}$
with neighbor classes of the first class ($i=0$) of the microstructure at
t=$0\text{\,}\mathrm{s}$.

The four contact probability definitions described by eq. (19) are defined for
the neighborhood of the first grain class of the microstructure at the initial
time t=$0\text{\,}\mathrm{s}$, as plotted in fig. 10. The probabilities are
used in the context of Maire et al. model detailed in section 2.4 by modifying
eq. (13). In eq. (19) with $m=0$, the contact probability $p_{(i,j)}^{0}$ is
computed in terms of number with no influence of the grain size. In fig. 10,
the orange curve shows that the majority of the weight is given to smaller
neighbor grains as only the frequency of occurrence in the microstructure is
taken into account. The $p_{(i,j)}^{1}$ formulation uses the perimeter of the
grain class to describe the contact probability. This latter is taking the
highest probability values on the first one-third grain classes, which
traduces that small grain classes are still more represented in the
neighborhood of the first class in this description. $p_{(i,j)}^{2}$
formulation considers the surface of the neighbor grain class $j$ to build the
contact probability. A shift toward the middle grain sizes is observed,
leading to larger grains taking part of the neighborhood compared to
$p_{(i,j)}^{0}$ and $p_{(i,j)}^{1}$ definitions. Finally, the description
$p_{(i,j)}^{3}$ imposes a neighborhood based on the volume of the neighbor
grain classes which gives more weights to larger grain classes as highlighted
by the green curve in fig. 10. Looking solely at neighborhood construction,
none of these representations seems more justified than others. Depending on
the selected representation, an emphasis on certain groups of grains is done
as explained above.

##### Impact on the distribution results

To select one of the above contact probability definitions for the study,
grain size distribution results will be compared using a GG test case. For the
three investigated temperatures, 1000, 1050 and 1100°C an annealing of two
hours is simulated with the four contact probabilities. The GSD results are
compared to the data converted into a 3D GSD in fig. 11 using the Saltykov
method described in section 3.2. The experimental data are represented by the
discrete black histogram. At all three temperatures, $p_{(i,j)}^{3}$ seems to
provide a better fit of the tail of the distribution. This volume-based
description brought by eq. (19) with $m=3$ accentuates the topological effect
of the neighborhood construction by giving more weight to bigger grains. If a
homogeneous microstructure is composed solely of small grains, the contact
probability description will give similar contact probabilities to these
grains with a comparable volume. However, if a heterogeneous microstructure
with large and small grains is considered, the volumic probability will bring
a topological aspect by giving more weight to larger grains. Indeed, if bigger
grains have a better representation in the neighborhood of a grain class, then
the $dV$ exchange achieved with the GB migration eq. (9) for these grain
classes is increased. Therefore, they have statistically more chances to grow
and be represented in the distribution in further time steps.

(a) $1000\text{\,}\mathrm{\SIUnitSymbolCelsius}$

(b) $1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$

(c) $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$

Figure 11: Comparison of the impact on the distribution of the $p_{(i,j)}^{m}$
with $m\in\llbracket{0,3}\rrbracket$ for an annealing of
$2\text{\,}\mathrm{h}$ at different temperatures for 316L.

####  4.1.3 Impact of the grain classes order in the neighborhood construction

The complexity of the neighborhood construction proposed by Maire et al. is
influenced by the selecting order of the grain classes in the GSD. As
mentioned in section 2.4, in Maire et al. original work [13] the ascending
order has been arbitrarily selected for the GSD. For Hillert and Abbruzzese et
al. models, the MGS evolution with respect to time shows no changes no matter
in which order the grain classes are selected, as no topology is involved in
the computation of their surrounding media.

To study the effect of the selecting order of the grain classes in the
neighborhood construction on the results, two other types of selecting order
are considered: GSD is either selected by decreasing grain sizes or randomly.

In fig. 12(a), the MGS evolution is strongly affected by the selection in
descending sorting order for construction of the neigborhood as described by
the orange solid line. The shuffle order of construction has a smaller impact
on the MGS kinetic compared to the latter one. The reduced mobility values
need to be re-identified to retrieve a good fit with the experimental data.
The values for these specific selecting orders of construction in Maire et al.
model are gathered in table 5 and the associated MGS evolutions are
represented by the dashed lines in fig. 12(a). The corresponding GSDs are
presented in terms of number frequency and volume fraction in fig. 12(b) and
12(c). When the reduced mobility is re-identified, predicted GSDs remain close
to each other independently of the selected sorting order of construction.
Ascending and descending construction orders tend to predict longer
distribution tails than the shuffle order. However, if the mobility is not re-
identified, the strong impact for MGS prediction can be explained by the
difference in neighborhood construction for the same current grain class. The
microstructure representations of these construction orders are schematically
presented in fig. 13(a) and fig. 13(b), describing two different construction
patterns. In ascending sorting order, bigger grains of the microstructure are
red grain classes, i.e. the ones that will not take part in the neighborhood
of the current class $i$. On the contrary, in the descending sorting order
construction, red grain classes are the smaller ones. This will modify the GB
migration volume exchanges between neighboring grains during GB migration.

For the forthcoming comparisons, the original ascending GSD sorting order will
be conserved, as in this case the reduced mobility value has a similar order
of magnitude to the one of the other models.

(a)

(b)

(c)

Figure 12: Comparing different selecting orders for the neighborhood
construction using the Maire et al. model. Test case selected here is a
$1\text{\,}\mathrm{h}$ annealing at
$1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$. (a) MGS evolution with respect
to time, GSD at the end of the heat treatment (t=$1\text{\,}\mathrm{h}$)
considering (b) number frequency and (c) volume fraction.

(a)

(b)

Figure 13: Specific neighborhood construction considering (a) the ascending and (b) the descending selecting order in the GSD.  Sorting order | Ascending | Descending | Shuffle
---|---|---|---
$M_{GB}\gamma_{GB}$($\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$) | 2.19e-13 | 5.00e-13 | 2.30e-13
Table 5: Identified reduced mobility values for Maire et al. model for
different selecting order at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for
316L.

###  4.2 Comparison of mean-field models using different initial
microstructures

In this section, initial mono- and bimodal distributions will be used to
evaluate the impact of heterogeneities in the initial GSD. The monomodal test
case use the initial experimental GSD. Thermal treatment simulations will be
compared to experimental data. However, bimodal comparisons are confronted
only between themselves as no experimental data were available for this case.
In both analyses, Maire et al. model is employed with a volumic contact
probability as detailed in section 4.1.2 and with an initial number of grain
classes of 1000 as deduced from section 4.1.1. An ascending sorting order for
the input distribution is considered as originally used in previous Maire et
al. work [13].

####  4.2.1 Comparison of mean-field models with a monomodal initial
microstructure

Fig. 14 (a) to (h) illustrate mean-field GSDs predictions after different
annealing times at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in comparison
with experimental data obtained thanks to EBSD. The tail of GSDs representing
larger grains is better predicted by Maire et al. model either considering
frequency in number or volume fraction representation of the GSD. However, the
volume fraction histograms show that none of the three models catches the
entire experimental distribution tail for any of the studied annealing times.
This may be due to both implementation hypotheses or also to experimental GSD
statistical representativity. Hillert model seems to better predict number
frequency histogram and under-estimate volumic predictions. However, for
Abbruzzese et al. and Maire et al. models, the first part of the GSD, composed
of small grain classes, shows good accordance with experimental data in volume
fraction (fig. 14 (a), (c), (e), (g)). On the other hand, the number frequency
predictions exposes some divergence with respect to the experimental data for
small grain sizes.

(a)  1h at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

(b)  1h at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

(c)  2h at at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

(d)  2h at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

(e)  3h at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

(f) 3h at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$

(g)  5h at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

(h)  5h at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$

Figure 14: GSDs comparisons in (a,c,e,g) number frequency and (b,d,f,h) in
volume fraction for Hillert, Abbruzzese et al. and Maire et al. models with
experimental data at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for (a,b)
1h, (c,d) 2h, (e,f) 3h and (g,h) 5h of thermal treatment on 316L.

One common strong hypothesis in models is considering spherical grains and
spherical evolution with capillarity. Models also make an important
approximation by considering grain boundaries properties as isotropic. Indeed,
grain boundaries energy $\gamma_{GB}$ and their mobility $M_{GB}$ are both
considered constant and identical for all grain classes used in the
simulation. However, experimental microstructures have been proven [24, 25],
to expose a dispersion in values of these properties. These hypotheses tend to
smooth all microstructure heterogeneities at the beginning or during
simulations, which may explains GSD extremity differences. The radius
variation exchange considered in the GB migration equation for the different
microstructure descriptions also impact the GSD prediction. Indeed, Hillert
model by employing an HEM account for these variations through the choice of
the MGS equation.

Experimental factors can also play a role in these differences. The
statistical representativity of experimental data can be limited and the
inverse Saltykov method is not a deterministic one. As a result no perfect
reproducibility of the data is possible. This can explain the difficulty of
models to predict GSD tail. The discussed models provide a general tendency of
the GSD evolution depending on the thermal conditions. Moreover, no special
treatment has been done to consider twining in this work even if these special
GB can be considered as partially taken into account in identified parameters
of the models such as $M_{GB}$.

For more quantified comparisons, $L^{2}(t)$ error is computed with eq. (18),
with $S_{i}$ (resp. $S_{i}^{\prime}$) corresponding to the $i^{th}$ bin grain
class area of the simulated GSD at time $t$ (resp. of the experimental GSD at
time $t$). Fig. 15(a) and 15(b) illustrate the comparison of the latter
criterion for the four experimented times in both representations in number
and volume fraction. High $L^{2}(t)$ error values can be explained by the
assumptions and experimental statistical representativity detailed above.
However, these results provide a good relative comparison basis between
models. As expected, Hillert model provide a good prediction rate in number
frequency. But, its volumic predictions are low compared to the two others.
Abbruzzese et al. and Maire et al. models provide a relatively constant
prediction in both representations. In order to qualify a model prediction
ability, good results in both representation are necessary. Maire et al.
expose here the overall lowest $L^{2}(t)$ values, making it the model with the
highest accuracy compare to the others.

(a)

(b)

Figure 15:  Comparison of $L^{2}(t)$ error computed from GSDs for the three
models and heat treatments from $1\text{\,}\mathrm{h}$ to
$5\text{\,}\mathrm{h}$ at $1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with a
description in (a) number frequency and (b) volume fraction.

In the case of GG, the simulation of an initial monomodal GSD is overall
similarly predicted by the three models. Maire et al. model improves GSD
predictions on larger grain sizes but Hillert gives satisfying results
considering the simplicity of its description when only GG mechanism is at
play.

####  4.2.2 Comparison of mean-field models on bimodal initial microstructure

By developing a test case using a bimodal initial GSD, the neighborhood
construction of the Maire et al. model is then tested for an initial
heterogeneous GSD. An input microstructure is selected with a MGS of
$42\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a four ATSM grain size
difference between the two selected grain populations to respect the bimodal
definition of the ASTM standard E112 [26] as shown in fig. 16(a). The
previously identified reduced mobility values are used. In this case, no
experimental data are available, thus only a relative comparison between
models will be made. The MGS evolution can be visually dissociated into two
kinetics. The first one is appearing in the range of the first
$10\text{\,}\min$ then a more steady increasing kinetic until the end of the
annealing treatment of $2\text{\,}\mathrm{h}$ is observed in fig. 16(b). Three
associated GSD at $5\text{\,}\min$, $10\text{\,}\min$, and
$2\text{\,}\mathrm{h}$ comparing the behavior of the models are given in fig.
16(c), 16(d) and 16(e). For all cases, there are very few distinctions between
models predictions and the bimodal aspect is rapidly smoothed after
$5\text{\,}\min$ with the apperance of a monomodal distribution, excepted
Hillert model predictions that retained a small bimodal distribution at
$5\text{\,}\min$ and $10\text{\,}\min$. The breaking point between the two
kinetics in the MGS evolution corresponds to the point where the monomodal
distribution is reached after about $10\text{\,}\min$.

(a)

(b)

(c)

(d)

(e)

Figure 16: Comparison of the GG predictions at
$1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ starting from an initial bimodal
distribution with a neighborhood construction selected in the ascending order
for the Maire et al. model: (a) Initial bimodal distribution, (b) MGS
evolution in time, and GSDs at (c) $5\text{\,}\min$, (d) $10\text{\,}\min$ and
(e) $2\text{\,}\mathrm{h}$.

##### Impact of the selecting order of neighborhood construction on the
distribution prediction

In the same way, as in section 4.1.3, an observation of the use of a
descending selecting order of neighborhood construction is performed.
Similarly, the MGS evolution of Hillert and Abbruzzese et al. in fig. 17(a)
remains unchanged and Maire et al. model kinetics is one more time slowed
down. However, the associated GSDs at $5\text{\,}\min$ and $10\text{\,}\min$
show well the bimodality. Maire et al. model seems to retard the smoothing
effect conducting to the final Gaussian distribution as shown in fig. 17(b)
and 17(c). Indeed, in both graphs, the small grain size population keeps a
higher number frequency in the case of Maire et al. model than for other GG
models. In addition to the construction impact due to the selecting order as
described in fig. 13, this construction technique does not allow the same
neighborhood diversity for all grain classes. As exposed by fig. 4, grains
classes selected in the middle of the construction process will benefit from a
more important number of neighbors as part of it is built with blue grains
having incomplete neighborhoods. As it gets to this end of the construction,
less grain classes are available to be part of the neighborhood. Considering
this point, the order of selection of neighborhood construction has an impact
on the GSD prediction even if the bijectivity is taken into account. This
topological information is consistent with what is observed on the GSD
evolution. In the bimodal case, it seems that the ascending order construction
promotes the elimination of the small grains population of the distribution
with a faster kinetic. Indeed, the curvature driving pressure of the GG
phenomenon will tend to facilitate the disappearance of smaller grains
considering the latter neighborhood construction. However, this elimination
kinetic seems to be postponed when the descending construction order
distribution is employed. In this case, small grains classes will be better
dispersed in the neighborhood construction and least favorable to
disappearance. This conducts to a higher MGS in the case of the ascending
order distribution around $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ while the
descending counterpart case barely reaches
$66\text{\,}\mathrm{\SIUnitSymbolMicro m}$. And finally, if an analysis of the
remaining grain classes number at the end of the first phase around
$1000\text{\,}\mathrm{s}$ of heat treatment is done, a difference of 90 grain
classes in the considered system is observed. Sorting order of the grain
classes using the specific neighborhood construction of the Maire et al. model
has a direct impact on microstructure evolution and therefore GSD predictions.

(a)

(b)

(c)

(d)

Figure 17: Comparison of the GG predictions at
$1100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ starting from an initial bimodal
distribution with a neighborhood construction selected in the descending order
for the Maire et al. model: (a) MGS evolution in time, and GSDs at (b)
$5\text{\,}\min$, (c) $10\text{\,}\min$ and (d) $2\text{\,}\mathrm{h}$.

##  5 Conclusion

Different GG mean-field models were investigated in this article by first a
detailed explanation of their equations/hypotheses and by comparing heat
treatment predictions on 316L steel. The neighborhood description in Maire et
al. model is based on the original work of Abbruzzese et al. [8] that
described a 2D GG model with a statistical neighborhood based on contact
probabilities involving the entire microstructure. It relies on a hybrid
description by using the statistical approach of contact probabilities to
define neighbors surfaces in contact with grain classes coupled with a
deterministic number of neighbors ruled by the use of a bijectivity criterion.
To optimize the accuracy of the discussed models, parameters analyses were
performed to observe their impact on the GSD predictions. First, a convergence
study was achieved to optimize the initial and final number of grain classes
in order to ensure statistical representativity. Then, the contact probability
definition was identified to be a leverage quantity in the GSD description.
Indeed, the latter impacts the distribution of the contact surface of neighbor
classes. Four contact probability have been tested, from a ratio in number not
involving the size of the grains to a volumic fraction. This means that for a
contact probability in number, the neighbor grains size will only impact the
grain evolution through its curvature radius in the GB migration. However, if
a volumic contact probability is considered, a more important weight will be
given to large grains which emphasizes the effect of grain topology. The
volumic contact probability has been selected for the previous reasons. In
particular, this has proven to give better GSD predictions by improving the
description of the tail of the GSD when compared to experimental data as shown
in section 4. And finally, the selected neighborhood construction order in the
GSD, impacts strongly the microstructure evolution in Maire et al. model. For
this model, reduced mobility values need to be re-identified in order to be
predictive when a different selecting order is adopted.

Once optimized parameters were determined, a focus was made on comparing
predicted GSD to EBSD experimental data for one to five hours annealing. To
enable the comparisons, the identification of reduced mobility with respect to
temperature and the use of the Saltykov method were necessary. Identified
values of the reduced mobility ensure that the models exhibit a similar MGS
evolution than the reference experimental data. On the other hand, Saltykov
method enables to convert 2D GSD histograms to 3D discrete GSD providing a way
for 2D to 3D conversion of experimental EBSD data. From these comparative
histograms, a good general accordance of all models with experimental data was
observed. However, Maire et al. model gives more satisfying results in
describing distribution tails than Hillert and Abbruzzese et al. ones.
$L^{2}(t)$ computation also gives a reduced error for Maire et al. model on
the studied GG cases. When comparing implementation simplicity and GSD
response, Hillert model gives good predictions with a simpler medium
description.

We have discussed in this article the interest of Maire et al. model solely in
the frame of GG. Originally designed to model discontinuous dynamic
recrystallization phenomenon [13], its strength relies on accounting for
topology in the considered grain neighborhoods. It is especially powerful when
dealing with different types of grains. By distinguishing RX grains from non-
RX ones, the specific neighborhood construction provides better predictions of
GSD with respect to experimental GSD. While this article was dedicated to
validate the neighborhood construction in the case of GG, new developments
aiming to improve GSD predictions during and after recrystallization is to be
followed.

## Acknowledgements

The authors thank ArcelorMittal, Aperam, Aubert & Duval, CEA, Constellium,
Framatome, Safran and Transvalor companies and the ANR for their support
through the DIGIMU consortium and RealIMotion ANR industrial Chair (Grant No.
ANR-22-CHIN-0003).

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# Limit points and additive group actions

Ivan Arzhantsev  HSE University, Faculty of Computer Science, Pokrovsky
Boulevard 11, Moscow, 109028 Russia<EMAIL_ADDRESS>
###### Abstract.

We show that an effective action of the one-dimensional torus
${\mathbb{G}}_{m}$ on a normal affine algebraic variety $X$ can be extended to
an effective action of a semi-direct product
${\mathbb{G}}_{m}\rightthreetimes{\mathbb{G}}_{a}$ with the same general orbit
closures if and only if there is a divisor $D$ on $X$ that consists of
${\mathbb{G}}_{m}$-fixed points. This result is applied to the study of orbits
of the automorphism group $\operatorname{Aut}(X)$ on $X$.

###### Key words and phrases:

Affine variety, torus action, additive group action, locally nilpotent
derivation

###### 2010 Mathematics Subject Classification:

Primary 14J50, 14R20; Secondary 13A50, 14L30

The research was supported by Russian Science Foundation, grant 19-11-00056

##  1\. Introduction

Let $X$ be an irreducible affine variety over an algebraically closed field
${\mathbb{K}}$ of characteristic zero. Assume that $X$ is equipped with an
effective action ${\mathbb{G}}_{m}\times X\to X$ of the one-dimensional
algebraic torus ${\mathbb{G}}_{m}$. If the orbit ${\mathbb{G}}_{m}\cdot x$ of
some point $x\in X$ is non-closed, the complement
$\overline{{\mathbb{G}}_{m}\cdot x}\setminus{\mathbb{G}}_{m}\cdot x$ consists
of one point $x_{0}$ called the _limit point_ of the orbit
${\mathbb{G}}_{m}\cdot x$.

Limit points play an important role in the theory of algebraic transformation
groups. Let $G$ be a reductive algebraic group acting linearly in a vector
space $V$. A vector $v\in V$ is called _nilpotent_ if the closure of the orbit
$G\cdot v$ contains zero. Equivalently, all homogeneous invariant polynomials
of positive degree on $V$ vanish at $v$. The Hilbert-Mumford Criterion proved
by Hilbert (1893) for the action of the unimodular group in a space of forms
and by Mumford (1965) in the general situation claims that a vector $v\in V$
is nilpotent if and only if there is a one-dimensional torus
${\mathbb{G}}_{m}\subseteq G$ such that $0\in\overline{{\mathbb{G}}_{m}\cdot
v}$; see e.g. [20, Theorem 5.1].

More generally, Richardson and Birkes proved that if a reductive algebraic
group acts regularly on an affine variety $X$ and $x\in X$ is some point, then
there is a one-dimensional torus ${\mathbb{G}}_{m}\subseteq G$ such that the
closure $\overline{{\mathbb{G}}_{m}\cdot x}$ intersects the (unique) closed
$G$-orbit in $\overline{G\cdot x}$; see [20, Theorem 6.9].

The study of limit points of a ${\mathbb{G}}_{m}$-action on a smooth
projective variety $X$ and the ${\mathbb{G}}_{m}$-module structure on the
tangent space of a limit point leads to the so-called Bialynicki-Birula
decomposition of the variety $X$ into locally closed
${\mathbb{G}}_{m}$-invariant subsets [8]. This decomposition is used in many
problems of Algebraic Geometry, Topology and Representation Theory.

One may ask whether the limit point $x_{0}$ of a non-closed orbit
${\mathbb{G}}_{m}\cdot x$ on an affine variety $X$ is equivalent to the point
$x$ in the sense that $x_{0}$ can be sent to $x$ by an automorphism of $X$. In
general, this is not the case at least because the point $x_{0}$ can be
singular while $x$ is smooth. At the same time, the cases when $x_{0}$ is
equivalent to $x$ are of particular interest, and they are the object of study
of this note.

One way to guarantee that $x_{0}$ and $x$ are equivalent is to find an
algebraic group $H$ acting on $X$ such that the orbit $H\cdot x$ contains both
the orbit ${\mathbb{G}}_{m}\cdot x$ and the point $x_{0}$. We are going to
apply this idea in the case when $H$ is the additive group
${\mathbb{G}}_{a}=({\mathbb{K}},+)$.

######  Definition 1.

Let $X$ be an irreducible affine variety with an efffective action of the one-
dimensional torus ${\mathbb{G}}_{m}$. We say that a regular action
${\mathbb{G}}_{a}\times X\to X$ is _compatible_ if the subgroup
${\mathbb{G}}_{m}$ normalizes the subgroup ${\mathbb{G}}_{a}$ in the
automorphsim group $\operatorname{Aut}(X)$ and generic
${\mathbb{G}}_{a}$-orbits on $X$ coincide with closures of generic
${\mathbb{G}}_{m}$-orbits.

Equivalently, to construct a compatible ${\mathbb{G}}_{a}$-action on an affine
${\mathbb{G}}_{m}$-variety $X$ is the same as to extend the
${\mathbb{G}}_{m}$-action to an effective action of a semi-direct product
${\mathbb{G}}_{m}\rightthreetimes{\mathbb{G}}_{a}$ such that the generic orbit
closures of the extended action coincides with generic orbit closures of the
${\mathbb{G}}_{m}$-action.

Let us recall that any orbit of a unipotent group action on an affine variety
is closed [20, Section 1.3]. So generic ${\mathbb{G}}_{a}$-orbits on $X$ are
closed curves isomorphic to the affine line.

Now we come to the main result of this note.

######  Theorem 1.

Let $X$ be a normal affine variety with an effective action of the one-
dimensional torus ${\mathbb{G}}_{m}$. Then there exists a compatible
${\mathbb{G}}_{a}$-action on $X$ if and only if the variety $X$ contains a
prime divisor $D$ that is fixed by ${\mathbb{G}}_{m}$ pointwise.

Theorem 1 allows for several applications. We say that an algebraic variety
$X$ is _almost homogeneous_ if there is a regular action $G\times X\to X$ of
an algebraic group $G$ with an open orbit.

######  Corollary 1.

Let $X$ be a normal affine almost homogeneous variety and $D$ be a prime
divisor on $X$. Assume that there is a non-trivial action of a one-dimensional
torus ${\mathbb{G}}_{m}$ on $X$ such that $D$ is fixed by ${\mathbb{G}}_{m}$
pointwise. Then the automorphism group $\operatorname{Aut}(X)$ acts on $X$
with an open orbit and this orbit intersects the divisor $D$.

Let us recall that an affine variety $X$ is called _rigid_ , if $X$ admits no
non-trivial action of the group ${\mathbb{G}}_{a}$. It is well known that $X$
is rigid if and only if the algebra ${\mathbb{K}}[X]$ admits no non-zero
locally nilpotent derivation [14, Section 1.5.1].

######  Corollary 2.

Let $X$ be a normal affine variety with an effective action of the one-
dimensional torus ${\mathbb{G}}_{m}$. Assume that there is a prime divisor $D$
on $X$ that is fixed by ${\mathbb{G}}_{m}$ pointwise. Then the variety $X$ is
not rigid. In particular, if $\dim X\geqslant 2$ then the automorphism group
$\operatorname{Aut}(X)$ is not a (finite dimensional) algebraic group.

If an affine variety $X$ admits two actions of the torus ${\mathbb{G}}_{m}$
that do not commute, then $X$ admits a non-trivial ${\mathbb{G}}_{a}$-action;
see [12, Section 3] and [6, Proof of Theorem 2.1]. This result plays a key
role in the description of the automorphism group of a rigid variety [6].
Theorem 1 is also a result of the same type: it guarantees the existence of a
non-trivial ${\mathbb{G}}_{a}$-action on $X$ provided $X$ admits a
${\mathbb{G}}_{m}$-action of a specific form.

We prove Theorem 1 and Corollaries 1-2 in Section 2. In Section 3 we give
examples that illustrate Theorem 1. In fact, the case of root subgroups on
non-degenerate affine toric varieties [7, Section 2] was the starting point of
this work. We discuss this example in detail and present some generalizations.

Let us mention some more researches related to the subject of this paper. In
[13], a classification of locally nilpotent derivations on the algebra of
regular functions on an affine surface with a ${\mathbb{C}}^{*}$-action is
given. In particular, [13, Theorem 3.12] deals with locally nilpotent
derivations, which we need in the proof of Theorem 1, in the case of surfaces.
A related result on existence of ${\mathbb{G}}_{a}$-actions on affine
varieties equipped with an action of the torus ${\mathbb{G}}_{m}$ may be found
in [16, Proposition 3]. Finally, when the first version of this paper was
written we have learned that a result similar to Theorem 1 appeared
independently in the diploma paper by Valentin Losev (unpublished).

_Acknowledgments._ The author is grateful to Alvaro Liendo for a fruitful
discussion, and to Sergey Gaifullin, Michail Zaidenberg, and two anonymous
reviewers for many useful comments, suggestions and references to related
results.

##  2\. Proofs of main results

Let $X$ be an irreducible algebraic variety equipped with a regular action
$T\times X\to X$ of an algebraic torus $T$. Consider a regular action
${\mathbb{G}}_{a}\times X\to X$ such that the image of ${\mathbb{G}}_{a}$ in
the automorphism group $\operatorname{Aut}(X)$ is normalized by the image of
$T$. We say that such an action is _vertical_ if the induced action of the
group ${\mathbb{G}}_{a}$ on the field ${\mathbb{K}}(X)^{T}$ of rational
$T$-invariants is identical, and is _horizontal_ otherwise. Equivalently, a
$T$-normalized ${\mathbb{G}}_{a}$-action on $X$ is vertical if a generic
${\mathbb{G}}_{a}$-orbit on $X$ is contained in the closure of a $T$-orbit on
$X$ [18, Section 1.2]; it follows from Rosenlicht’s Theorem which claims that
generic orbits of an action of an algebraic group are separated by rational
invariants, see e.g. [20, Theorem 2.3]. In turn, horizontal
${\mathbb{G}}_{a}$-actions are characterized by the condition that generic
${\mathbb{G}}_{a}$-orbits are transversal to generic $T$-orbits. In
particular, compatible ${\mathbb{G}}_{a}$-actions defined in the introduction
are precisely vertical actions on affine varieties for $T={\mathbb{G}}_{m}$.

Let us consider ${\mathbb{G}}_{m}$-actions on affine varieties more
systematically. There is a natural correspondence between effective
${\mathbb{G}}_{m}$-actions on an affine variety $X$ and
${\mathbb{Z}}$-gradings on the algebra $A={\mathbb{K}}[X]$, i.e.

$A=\bigoplus_{i\in{\mathbb{Z}}}A_{i},$

such that the set of indices $i$ with $A_{i}\neq 0$ generates the group
${\mathbb{Z}}$. Let us recall that a homogeneous component $A_{i}$ consists of
functions $f\in{\mathbb{K}}[X]$ such that $f(t\cdot x)=t^{i}f(x)$ for all
$t\in{\mathbb{G}}_{m}$ and all $x\in X$. Up to automorphism of the torus
${\mathbb{G}}_{m}$ we may assume that there exists a positive $i$ with
$A_{i}\neq 0$.

A ${\mathbb{G}}_{m}$-action on an affine variety $X$ is called _elliptic_ if
the only closed orbit is a ${\mathbb{G}}_{m}$-fixed point $P$. In this case
all ${\mathbb{G}}_{m}$-orbits in $X$ contain $P$ in their closures.
Equivalently, the algebra of invariants ${\mathbb{K}}[X]^{{\mathbb{G}}_{m}}$
coincides with the ground field ${\mathbb{K}}$. In terms of gradings, elliptic
actions correspond to positive gradings $A=\oplus_{i\geqslant 0}A_{i}$ of the
algebra $A={\mathbb{K}}[X]$ with $A_{0}={\mathbb{K}}$.

Oppositely, an effective ${\mathbb{G}}_{m}$-action is _hyperbolic_ if generic
${\mathbb{G}}_{m}$-orbits on $X$ are closed. This means that the grading
$A=\oplus_{i\in{\mathbb{Z}}}A_{i}$ includes non-zero components of both
positive and negative degrees. Moreover, this condition is equivalent to
existence of at least one closed orbit of positive dimension. In particular,
if $A=\oplus_{i\geqslant 0}A_{i}$ then the closure of any one-dimensional
${\mathbb{G}}_{m}$-orbit contains a limit point.

Finally, we say that an effective ${\mathbb{G}}_{m}$-action on an irreducible
affine variety $X$ is _parabolic_ if there is an open subset $U\subseteq X$
such that all ${\mathbb{G}}_{m}$-orbits in $U$ are non-closed in $X$ and their
closures in $X$ do not intersect pairwise.

######  Lemma 1.

Let $X$ be a normal affine variety equipped with an effective action of the
torus ${\mathbb{G}}_{m}$. The following conditions are equivalent.

  1. a)

The action is parabolic.

  2. b)

There is a prime divisor $D$ in $X$ that is fixed by ${\mathbb{G}}_{m}$
pointwise.

  3. c)

The grading has the form $A=\oplus_{i\geqslant 0}A_{i}$ and the transcendence
degree of the subalgebra $A_{0}={\mathbb{K}}[X]^{{\mathbb{G}}_{m}}$ equals
$\dim X-1$.

Under these conditions, the divisor $D$ coincides with the set of
${\mathbb{G}}_{m}$-fixed points on $X$.

###### Proof.

Let us consider the quotient morphism $\pi\colon X\to
Y=X/\\!/{\mathbb{G}}_{m}:=\operatorname{Spec}{\mathbb{K}}[X]^{{\mathbb{G}}_{m}}$.
This morphism separates closed orbits and sends ${\mathbb{G}}_{m}$-invariant
closed subsets of $X$ to closed subsets of $Y$ [20, Section 4.4]. Since every
fiber of $\pi$ contains a unique closed orbit, all fibers are connected. By
[2, Proposition 4] we conclude that generic fibers of $\pi$ are irreducible.

We show that a) implies b). Since the action is parabolic, generic orbit
closures on $X$ are separated by the morphism $\pi$. This proves that generic
fibers of $\pi$ are one-dimensional and $\dim Y=\dim X-1$. Moreover, the
subvariety of ${\mathbb{G}}_{m}$-fixed points on $X$ projects to $Y$
dominantly. We conclude that some irreducible component $D$ of this subvariety
is a prime divisor.

Now let us check that b) implies c). Since the ${\mathbb{G}}_{m}$-action on
$X$ is effective we have $\dim Y<\dim X$. On the other hand, the divisor $D$
projects to a closed subset of $Y$ and the projection is injective. This means
the restriction of $\pi$ to $D$ is bijective. In particular, all closed orbits
on $X$ are fixed points and $\dim Y=\dim D=\dim X-1$. The first condition
means that $A_{i}=0$ for all $i<0$ and the second one means that the
transcendence degree of $A_{0}$ is $\dim X-1$.

Let us show that c) implies a). Condition c) means that $\dim Y=\dim X-1$ and
all closed orbits in $X$ are fixed points. This implies that generic orbits on
$X$ are non-closed and their closures coincide with generic fibers of $\pi$.
We conclude that generic orbit closures do not intersect pairwise, i.e., the
action is parabolic.

Finally, since the restriction of $\pi$ to $D$ is bijective, every closed
orbit on $X$ is a point in $D$. ∎

###### Proof of Theorem 1.

Assume first that there is a compatible ${\mathbb{G}}_{a}$-action on $X$. Then
closures of generic ${\mathbb{G}}_{m}$-orbits on $X$ coincide with
${\mathbb{G}}_{a}$-orbits and so they do not intersect pairwise. This means
that the ${\mathbb{G}}_{m}$-action is parabolic and implication from a) to b)
in Lemma 1 concludes the proof.

Now let us assume that there is a prime divisor $D$ on $X$ that consists of
${\mathbb{G}}_{m}$-fixed points. Then we have a grading $A=\oplus_{i\geqslant
0}A_{i}$ as in Lemma 1, c). Let us show that all weight components $A_{s}$,
$s>0$ are non-zero. Assume that some $A_{s}=0$. We localize the algebra $A$ by
all non-zero elements of the subalgebra $B:=A_{0}$. The new algebra
$C=A_{(B)}$ inherits the grading $C=\oplus_{i\geqslant 0}C_{i}$ from $A$, and
$A_{s}=0$ implies $C_{s}=0$. By Lemma 1, a), elements of $B$ separate generic
${\mathbb{G}}_{m}$-orbits on $X$. This implies that the field
$K:={\mathbb{K}}(X)^{{\mathbb{G}}_{m}}$ of rational invariants coincides with
the field of fractions of the algebra $B$ [20, Lemma 2.1].

By construction, any two homogeneous elements in $C$ of the same degree are
proportional over $K$. This shows that any homogeneous component in $C$ is at
most one-dimensional over $K$. Since $C$ is a domain, we conclude that $C$ is
a semigroup algebra $K[\Gamma]$ for some subsemigroup $\Gamma$ in
$({\mathbb{Z}}_{\geqslant 0},+)$. By assumption, the algebra $A$ is integrally
closed in its field of fractions. This implies that the algebra $C$ is
integrally closed as well. But then the algebra $C$ is isomorphic to the
polynomial algebra $K[T]$, where the element $T$ has degree $d>0$. Since the
action of ${\mathbb{G}}_{m}$ on $X$ is effective, we have $d=1$ and so
$C_{s}\neq 0$ for all $s\geqslant 0$.

We are going to construct a locally nilpotent derivation $\delta$ of the
algebra $A$ that is homogeneous with respect to the grading and such that
$\operatorname{Ker}\delta=B$. We start with the algebra $C=K[T]$ and the
derivation $\delta^{\prime}=\frac{\partial}{\partial T}$. Let
$a_{1},\ldots,a_{m}$ be homogeneous generators of the algebra $A$. Then there
is an element $b\in B$ such that all elements
$b\delta^{\prime}(a_{1}),\ldots,b\delta^{\prime}(a_{m})$ are in $A$. Then the
derivation $\delta=b\delta^{\prime}$ is the required locally nilpotent
derivation of the algebra $A$. Consider the ${\mathbb{G}}_{a}$-action of $X$
induced by the group $\\{\text{exp}(s\delta)\,|\,s\in{\mathbb{K}}\\}$ of
automorphisms of the algebra $A$. Such a ${\mathbb{G}}_{a}$-subgroup is
normalized by ${\mathbb{G}}_{m}$ because the derivation $\delta$ is
homogeneous, and the ${\mathbb{G}}_{a}$-action is vertical because by
construction $\delta$ annihilates all elements in
${\mathbb{K}}(X)^{{\mathbb{G}}_{m}}$. We conclude that the constructed
${\mathbb{G}}_{a}$-action is compatible. ∎

######  Remark 1.

The inverse implication in Theorem 1 should also follow from [18, Corollary
2.8]. This approach is based on the technique of proper polyhedral divisors of
Altmann and Hausen [1]. In this note we prefer to use more elementary
arguments given above.

###### Proof of Corollary 1.

Since $X$ is almost homogeneous, there is a regular action $G\times X\to X$ of
an algebraic group $G$ with an open orbit $\mathcal{V}$. Then the union of
shifts $s\cdot\mathcal{V}$, where $s$ runs through all automorphisms of $X$,
is an open orbit $\mathcal{W}$ of the group $\operatorname{Aut}(X)$ on $X$. By
Theorem 1, there is a compatible ${\mathbb{G}}_{a}$-action on $X$. Since the
divisor $D$ coincides with the set of ${\mathbb{G}}_{m}$-fixed points on $X$,
generic ${\mathbb{G}}_{a}$-orbits intersect both $\mathcal{V}$ and $D$. This
shows that generic points on $D$ lie in the same orbit of
$\operatorname{Aut}(X)$ as $\mathcal{V}$. So the orbit $\mathcal{W}$
intersects the divisor $D$. ∎

###### Proof of Corollary 2.

By Theorem 1, there is a compatible ${\mathbb{G}}_{a}$-action on $X$. So the
variety $X$ is not rigid. It is well known that if an affine variety $X$ of
dimension at least $2$ admits a ${\mathbb{G}}_{a}$-action, then the
automorphism group $\operatorname{Aut}(X)$ is not a (finite dimensional)
algebraic group; see e.g. [17, Proposition 7.5]. Let us recall the arguments
for this. Let $\delta$ be a locally nilpotent derivation on the algebra
${\mathbb{K}}[X]$ that corresponds to the compatible
${\mathbb{G}}_{a}$-action. The kernel $\operatorname{Ker}\delta$ is a
subalgebra of transcendence degree $\dim X-1$; see [14, Principle 11]. In
particular, the kernel $\operatorname{Ker}\delta$ is an infinite dimensional
${\mathbb{K}}$-vector space. For any finite dimensional subspace
$V\subseteq\operatorname{Ker}\delta$ one can consider the subgroup
$\\{\text{exp}(f\delta)\,|\,f\in V\\}$ of the automorphism group
$\operatorname{Aut}(X)$. This is a commutative unipotent group of dimension
$\dim V$. Since $\dim V$ is unbounded, all such subgroups can not be contained
in a finite dimensional algebraic group. ∎

######  Remark 2.

Both Theorem 1 and Corollaries 1-2 do not hold if the variety $X$ is non-
normal: one may consider the cuspidal cubic $V(y^{2}-x^{3})$ in
${\mathbb{A}}^{2}$ with the torus action $t\cdot(x,y)=(t^{2}x,t^{3}y)$.

##  3\. Examples and applications

###  3.1. Subtorus actions

Let $N$ be the lattice of one-parameter subgroups of an algebraic torus $T$.
Denote by $M$ the dual lattice of characters of $T$, and let
$\langle\cdot,\cdot\rangle\colon N\times M\to{\mathbb{Z}}$ be the pairing
between $N$ and $M$. We consider a regular action of $T$ on a normal affine
variety $X$ and the corresponding grading

$A=\bigoplus_{u\in M}A_{u}$

of the affine algebra $A={\mathbb{K}}[X]$. Let
$N_{{\mathbb{Q}}}=N\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ and
$M_{{\mathbb{Q}}}=M\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ be the rational vector
spaces generated by $N$ and $M$, respectively. We define the _weight monoid_
$M_{X}$ of the affine $T$-variety $X$ as the set of weights $u\in M$ such that
$A_{u}\neq 0$, and the _weight cone_ $\omega(X)$ as the cone in
$M_{{\mathbb{Q}}}$ generated by the monoid $M_{X}$. Since the algebra $A$ is
finitely generated, the monoid $M_{X}$ is finitely generated and the cone
$\omega(X)$ is polyhedral.

We say that a one-dimensional subtorus ${\mathbb{G}}_{m}$ in $T$ is
_straightening_ if the induced ${\mathbb{G}}_{m}$-action on $X$ is parabolic.
Let us recall that a _facet_ of a polyhedral cone is a proper face of maximal
dimension.

######  Proposition 1.

Let $X$ be a normal affine $T$-variety. Then straightening subtori in $T$ are
in bijection with some facets of the cone $\omega(X)$. In particular, the
number of straightening subtori in $T$ is at most finite.

###### Proof.

Observe that a one-parameter subgroup ${\mathbb{G}}_{m}$ in $T$ is nothing but
an integer linear function $l$ on the space $M_{{\mathbb{Q}}}$. In these
terms, the grading on $A$ corresponding to the ${\mathbb{G}}_{m}$-action on
$X$ is

$A=\bigoplus_{i\in{\mathbb{Z}}}A_{i},\quad\text{where}\quad
A_{i}=\bigoplus_{l(u)=i}A_{u}.$ (1)

We conclude that such a grading contains no negative component if and only if
the intersection of $\omega(X)$ with the hyperplane $\\{l=0\\}$ is a face of
$\omega(X)$. For any face $\tau$ of the cone $\omega(X)$ we denote by
$A(\tau)$ the subalgebra $\oplus_{u\in\tau}A_{u}$. If a face $\tau$ is a facet
of a face $\lambda$, then $A(\tau)\subseteq A(\lambda)$ and the transcendence
degree of $A(\lambda)$ exceeds the transcendence degree of $A(\tau)$ by at
least one. So for grading (1) without negative components we may have that the
transcendence degree of the subalgebra $A_{0}$ is $\dim X-1$ only if $A_{0}$
has the form $A(\lambda)$ for some facet $\lambda$ of the cone $\omega(X)$.
Lemma 1, c) implies that straightening one-parameter subgroups in $T$
correspond to some facets of $\omega(X)$. ∎

Now we come to an important particular case of subtorus actions.

###  3.2. Affine toric varieties

Assume that a torus $T$ acts effectively on a normal affine variety $X$ with
an open orbit, i.e., $X$ is an affine toric $T$-variety. Equivalently, we have
$\dim A_{u}=1$ for any $u\in M_{X}$. Moreover, in this case the algebra $A$ is
the semigroup algebra of the semigroup $M_{X}=\omega(X)\cap M$. We refer to
[10, 15, 19] for a general theory of toric varieties.

Under these assumptions, the transcendence degree of every subalgebra
$A(\lambda)$, where $\lambda$ is a facet of the cone $\omega(X)$, equals $\dim
X-1$. So we have a bijection between straightening subtori in $T$ and facets
of $\omega(X)$.

Let $\sigma(X):=\omega(X)^{\vee}$ be the cone dual to $\omega(X)$ in the dual
space $N_{{\mathbb{Q}}}$. Then the primitive lattice vectors
$p_{1},\ldots,p_{m}$ on the rays of $\sigma(X)$ are precisely the
straightening subgroups in $T$. It is well known that the rays of $\sigma(X)$
are in bijection with $T$-invariant prime divisors $D_{1},\ldots,D_{m}$ on
$X$; see, e.g., [10, Theorem 3.2.6]. Under this correspondence, the divisor
$D_{i}$ is the set of fixed points of the subgroup $p_{i}$ [10, Proposition
3.2.2].

We say that a vector $e\in M$ is a _Demazure root_ of the cone $\sigma(X)$ if
there is $1\leqslant s\leqslant m$ such that $\langle p_{s},e\rangle=-1$ and
$\langle p_{i},e\rangle\geqslant 0$ for all $i\neq s$. So, the set
$\mathfrak{R}$ of all Demazure roots of the cone $\sigma(X)$ is a disjoint
union of subsets $\mathfrak{R}_{1},\ldots,\mathfrak{R}_{m}$ indexed by the
corresponding index $s$. One can easily check that if $\dim X\geqslant 2$ then
every set $\mathfrak{R}_{s}$ is infinite.

It is known that Demazure roots of $\sigma(X)$ are in bijection with
${\mathbb{G}}_{a}$-subgroups in $\operatorname{Aut}(X)$ normalized by $T$ [18,
Theorem 1.6]. Moreover, the ${\mathbb{G}}_{a}$-subgroup $H_{e}$ corresponding
to a root $e\in\mathfrak{R}_{s}$ acts on $X$ compatibly with the
${\mathbb{G}}_{m}$-subgroup given by the vector $p_{s}\in N$ [7, Proposition
2.1]. Summarizing these results, we obtain the following proposition.

######  Proposition 2.

Let $X$ be an affine toric $T$-variety. Then straightening
${\mathbb{G}}_{m}$-subgroups in $T$ are in bijection with $T$-invariant prime
divisors, and every $T$-invariant prime divisor on $X$ is the set of
${\mathbb{G}}_{m}$-fixed points for the corresponding straightening subtorus.

This shows that an open orbit $\mathcal{O}$ of the group
$\operatorname{Aut}(X)$ on $X$ intersects any $T$-invariant prime divisor. In
fact, it is known that $\mathcal{O}$ coincides with the regular locus of $X$,
and if $X$ is non-degenerate and $\dim X\geqslant 2$, then the group
$\operatorname{Aut}(X)$ acts on $\mathcal{O}$ infinitely transitively [7,
Theorem 2.1].

######  Remark 3.

On a non-normal affine toric variety $X$, an open
$\operatorname{Aut}(X)$-orbit may not intersect any $T$-invariant prime
divisor on $X$. This is the case, for example, for the Neil’s parabola from
Remark 2; for a general result on this effect, see [9, Theorem 2].

######  Remark 4.

A description of all orbits of the group $\operatorname{Aut}(X)$ on an affine
toric variety $X$ is obtained in [4, Theorem 5.1 and Proposition 5.6].

One should note that if we go from the toric case to an arbitrary torus action
on a normal affine variety, the situation may change completely. Let us
consider an example of a torus action with generic orbits of codimension one.
Fix positive integers $n_{0},n_{1},n_{2}$ with $n=n_{0}+n_{1}+n_{2}$. For each
$i=0,1,2$, fix a tuple $l_{i}\in{\mathbb{Z}}_{>0}^{n_{i}}$ with
$l_{i1}n_{i}>1$ and define a monomial

$T_{i}^{l_{i}}:=T_{i1}^{l_{i1}}\ldots
T_{in_{i}}^{l_{in_{i}}}\in{\mathbb{K}}[T_{ij};\ i=0,1,2,\,j=1,\ldots,n_{i}].$

By a trinomial we mean a polynomial of the form
$f=T_{0}^{l_{0}}+T_{1}^{l_{1}}+T_{2}^{l_{2}}$. A _trinomial hypersurface_ $X$
is the zero set $\\{f=0\\}$ in the affine space ${\mathbb{A}}^{n}$. One can
check that $X$ is a normal affine variety of dimension $n-1$ and the variety
$X$ carries a natural effective action of an $(n-2)$-dimensional torus $T$;
see [11]. By [3, Lemma 2], there is no vertical ${\mathbb{G}}_{a}$-action on
$X$. In particular, there is no $T$-normalized ${\mathbb{G}}_{a}$-action on
$X$ that is compatible with a one-dimensional subtorus in $T$.

###  3.3. Homogeneous fiber spaces

We are going to apply the results discussed in the previous subsection to a
wider class of almost homogeneous varieties. Let $G$ be a linear algebraic
group and $T$ be a subtorus in $G$. We take an affine toric $T$-variety $Y$
and consider the homogeneous fiber space $X=G*_{T}Y$, which is the quotient of
the direct product $G\times Y$ by the $T$-action $t\cdot(g,y)=(gt^{-1},ty)$;
see [20, Section 4.8] for details. The variety $G*_{T}Y$ projects to the
homogeneous space $G/T$, and all fibers of the projection are isomorphic to
$Y$. The $G$-action on $G\times Y$ by left multiplication on the first factor
induces the $G$-action on $X$ with an open orbit. Moreover, the $T$-action on
$G\times Y$ via the action on the second factor descents to a $T$-action on
$X$.

######  Proposition 3.

Let $G$ be a connected linear algebraic group, $T$ be a subtorus in $G$, and
$Y$ be an affine toric $T$-variety. Consider the homogeneous fiber space
$X=G*_{T}Y$. Then an open orbit of the group $\operatorname{Aut}(X)$ on $X$
intersects every $G$-invariant prime divisor on $X$.

###### Proof.

Every $G$-invariant prime divisor on $X$ has the form $G*_{T}D$, where $D$ is
a $T$-invariant prime divisor on $Y$. The action of a straightening
${\mathbb{G}}_{m}$-subgroup on $Y$ that fixes $D$ pointwise extends to $X$ and
has the divisor $G*_{T}D$ as the set of fixed points. So the assertion follows
from Corollary 1. ∎

###  3.4. Affine varieties with two orbits

Finally, let us show that if one adds a homogeneous divisor to a homogeneous
space then the extended space is still homogeneous.

######  Proposition 4.

Let $G$ be a reductive group and $X$ be a normal affine $G$-variety consisting
of two $G$-orbits $\mathcal{O}_{1}$, $\mathcal{O}_{2}$ with
$\dim\mathcal{O}_{1}=\dim\mathcal{O}_{2}+1$. Then the variety $X$ is
$\operatorname{Aut}(X)$-homogeneous.

###### Proof.

Since a normal variety is smooth in codimension one, the variety $X$ is
smooth. Let $K$ be the stabilizer of a point in the orbit $\mathcal{O}_{2}$.
By [20, Theorem 6.7], the variety $X$ is isomorphic to $G*_{K}V$, where $V$ is
a $K$-module. The condition $\dim\mathcal{O}_{1}=\dim\mathcal{O}_{2}+1$
implies that $V$ is one-dimensional. So $K$ acts on $V$ by some character. The
action of ${\mathbb{G}}_{m}$ on $G\times V$ by scalar multiplication on the
second factor descents to $G*_{K}V$, and the divisor $\mathcal{O}_{2}$
consists of ${\mathbb{G}}_{m}$-fixed points. So the assertion follows from
Corollary 1. ∎

######  Remark 5.

If the group $G$ is semisimple, a more general version of Proposition 4 is
given in [5, Theorem 5.6].

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