# An alternate approach for estimating grain-growth kinetics

M Prabhakar  P G Kubendran Amos<EMAIL_ADDRESS>Theoretical Metallurgy Group,
Department of Metallurgical and Materials Engineering,
National Institute of Technology Tiruchirappalli,
Tamil Nadu, India  Institute of Applied Materials (IAM-MMS), Karlsruhe
Institute of Technology (KIT),
Strasse am Forum 7, 76131 Karlsruhe, Germany

###### Abstract

Rate of grain growth, which aides in achieving desired properties in
polycrystalline materials, is conventionally estimated by measuring the size
of grains and tracking its change in micrographs reflecting the temporal
evolution. Techniques adopting this conventional approach demand an absolute
distinction between the grains and the interface separating them to yield an
accurate result. Edge-detection, segmentation and other deep-learning
algorithms are increasingly adopted to expose the network of boundaries and
the associated grains precisely. An alternate approach for measuring grain-
growth kinetics, that curtails the need for advanced image-processing
treatment, is presented in this work. Grain-growth rate in the current
technique is ascertained by counting the number of triple-( and quadruple-)
junctions, and monitoring its change during the microstructural evolution. The
shifted focus of this junction-based treatment minimises the significance of a
well-defined grain-boundary network, and consequently, the involvement of the
sophisticated techniques that expose them. A regression-based object-detection
algorithm is extended to realise, and count, the number of junctions in
polycrystalline microstructures. By examining the change in the number of
junctions with time, the growth rate is subsequently determined. Growth
kinetics estimated by the present junction-based approach, across a wide-range
of multiphase polycrystalline microstructures, agree convincingly with the
outcomes of the conventional treatment. Besides offering a novel technique for
grain-growth measurement, the analysis accompanying the current work unravels
a trend, compatible with the topological events, in the progressive evolution
of the triple-junctions count. The present approach, through its underlying
algorithm, provides a promising option for monitoring grain-growth during in-
situ investigations.

###### keywords:

Grain-Growth Kinetics, Triple Junctions, Quadruple Junctions, Multiphase
Polycrystalline Microstructures, Object Detections

##  1 Introduction

The unique, and often, striking properties of a nanocrystalline material are
primarily the outcome of the characteristic size of its constituent ‘crystals
’ or grains. In an application, wherein a nanocrystalline material is
employed, the span of its applicability is essentially the duration across
which the characteristic grain-size is retained [1]. In other words, under the
associated conditions, rate of grain growth determines the life of a
nanocrystalline component in any application. Sluggish the grain growth,
increased is the life span of nanocrystalline material [2]. This aspect of
nanomaterials becomes extremely critical in high temperature applications [3].
Besides these specific category of materials, properties of polycrystalline
systems, ranging from alloys to ceramics, are noticeably influenced by grain
size [4, 5]. Correspondingly, processing techniques are developed to establish
the desired grain size, ultimately yielding the required properties [6].
Designing of such processing techniques demands a quantitative understanding
of the grain-growth kinetics. Owing to these, and several other, critical
reasons different approaches have been developed and adopted to measure the
rate of grain growth.

The techniques employed to ascertain the rate of grain growth can broadly be
categorised into two. In the first approach, the kinetics are estimated by
measuring the average size of grains at specific instances during the temporal
evolution of polycrystalline microstructures. The techniques associated with
this approach can be referred to as ‘static ’ , since the growth rate is
determined from the temporally-fixed microstructures (images) captured at
different instances. Two well-known example of the static approaches are line-
intercept [7] and planimetric methods [8]. A line of specific length is
introduced in a polycrystalline micrograph, in the line-intercept method, and
the number of intersections between the grain boundaries and the line is
counted. In essence, finer the grains, larger the number of intersections. By
taking the reciprocal of the number of intersections, the intercept length is
calculated. A statistically more rigorous measure, called average intercept
length ($l(t)$), is ascertained from several parallel lines which is
subsequently translated to ASTM grain-size number [9]. While following the
similar approach of gathering relevant information from static representation
of microstructure, planimetric method estimates the grain size by focusing on
the number of complete grains encapsulated in test section of a definite area
[10]. The average grains per unit area ($N_{A}$) is estimated from several
sections of identical area across the polycrystalline micrograph. Through the
relation, $G(t)=3.322[\text{log}\leavevmode\nobreak\ N_{A}(t)]-2.954,$ the
average grain size, G(t), is calculated. Measuring the average grain size of
different polycrystalline micrographs, captured across the temporal evolution,
ultimately yields the growth kinetics in the static approach.

Owing to the straightforward framework, the static approaches are widely
adopted for measuring the average grain size in polycrystalline systems.
However, these techniques are rather arduous and demand scrupulous labor.
Furthermore, the accuracy of the outcome rendered by the static techniques
depends solely on the efficacy of etching. When the grain boundaries are not
sufficiently distinguished, the size measurement becomes rather compromised.
Besides, given its underlying framework, the static techniques cannot be
adopted to measure growth kinetics during in-situ observation. These
limitations of static methods have led to the development of the second
category of ‘dynamic ’ techniques to ascertain grain growth kinetics.

The dynamic approach characteristically measures the grain-growth kinetics
directly, as opposed to ascertaining it from the static representation of
micrographs captured at progressive instances. Correspondingly, the dynamic
techniques are fundamentally different and more sophisticated. One such
sophisticated dynamic method involves ultrasonic waves to measure the rate of
grain growth [11]. A material exposed to ultrasonic waves largely allows the
waves to pass through unimpeded. Any deviation from a propagation pathway is
observed only when the waves interact with an inhomogeneity. Polycrystalline
systems are fraught with grain boundaries that introduce inhomogeneities by
interrupting the specific alignment of atoms. Therefore, the scattering of
ultrasonic waves can be quantitatively related to the area of grain boundaries
in a material. Stated otherwise, the amount of ultrasonic waves scattered by a
fine-grained material is noticeably greater than a coarse-grained system.
Consequently, as the grains grow in a polycrystalline system, the scattering
of ultrasonic waves gradually reduces, and propagation becomes largely
unperturbed. Exploiting this mechanism, the growth kinetics of grains, in the
corresponding dynamic technique, is estimated by monitoring the rate of
progressive decrease in the scattering of the ultrasonic waves. Although this
method renders a real-time estimation of the grain-growth rate, it demands an
appropriate source for the invasive ultrasonic waves. The combined involvement
of the pulsed laser and the interferometer, which serve as the source and
detector of the ultrasonic waves, contributes to the intricacy of the
technique, thereby ultimately limiting its applicability.

In another dynamic approach, the change in enthalpy accompanying the grain
growth is measured and subsequently translated to the kinetics [12]. Grain
growth in the polycrystalline system is characterized by a progressive
decrease in the density of the grain boundaries. The increasing disappearance
of the sections of grain boundaries induces an exothermic change in the
interfacial enthalpy. This enthalpy change can be measured and tracked by
differential scanning calorimetry [13]. By monitoring the calorimetric signal,
which reflects the rate of enthalpy change, the kinetics of grain growth is
determined in this dynamic technique. The underlying principle of this
technique allows the measurement of growth kinetics across a given material,
in its entirety, as opposed to specific sections. However, this approach has a
telling limitation in its usage. The enthalpy change accompanying the
disappearance of interfaces is generally imperceptible and cannot be easily
detected. Consequently, the enthalpy-based method becomes viable only when the
change in the grain-boundary density is significant, as typically observed in
nanocrystalline materials.

Advancements made in artificial intelligence are increasingly employed to
understand the behavior of materials [14, 15, 16].  Some of these advancements
have recently been employed to aid in measuring grain-growth kinetics.
Considering that the inadequate realization of grain boundaries is a principal
source of inaccuracy in estimating grain size, and thereby growth rate, deep
learning techniques have been employed to aid the estimation of kinetics. In a
recent work, an edge detection algorithm is adopted to distinguish the
existing grain boundaries in a polycrystalline microstructure [17]. Upon
distinguishing the grain boundaries, the associated grains are realised as
closed contours by removing any discontinuities in the interface network and
removing any spurious edges. The static measurement techniques, in an
automated framework, are extended to the deep-learning-enhanced
polycrystalline micrographs for accurate measurement of the grain size.
Besides this approach, a technique estimating the grain size from the
intersection of the boundaries in the ‘topological-skeleton’ has subsequently
been reported [18]. Furthermore, a combination of object detection and
segmentation has been employed in a later study to estimate the grain size
[19]. These recent advancements essentially reinforce the preceding works
wherein an absolute distinction between the boundary, grains and any
additional phases is established largely by

  1. 1.

introducing a characteristic threshold for every microstructure to separate
the grains from the rest [20]

  2. 2.

employing a combination of segmentation with appropriate pre- and post-
processing techniques [21]

  3. 3.

a combination of segmentation and adaptive thresholding [22].

Highlights of the existing works are listed in Table 1.

Table 1: Chronological summary of the reported approaches .

Work | Technique | Grain size measurement
---|---|---
H. Peregrina-Barreto et al. (2013) [20] | Automatic Digital Sample Preparation (image processing) | Planimetric count
IA. Campbell et al. (2018) [21] | Watershed transform with pre- and post-processing | Aspect-ratio based estimation
S. Banerjee et al. (2019) [17] | Edge detection and closed contour formation | Intercept and Planimetric method
X. Li, L. Cui, J. Li et al. (2022) [18] | Topological-skeleton based algorithm | Intercept method
C. Gorynski et al. (2023) [19] | Mask R-CNN | Pixel-based (Mask) calculation

Irrespective of the differences in the microstructure processing, growth
kinetics in all the existing works are ascertained by first estimating the
accurate grain size through an established ‘static ’ technique and
subsequently tracking its evolution. Such framework for estimating growth
rate, though renders an accurate outcome which can be related to existing
standards, largely overlooks the evolution of other microstructural features
that accompany grain growth. Therefore, deviating from the principal focus on
grain size, and employing a suitable artificial intelligence (AI) tool, the
present work seemingly offers a novel approach for measuring grain-growth
kinetics.

Any polycrystalline microstructure encompasses following two characteristic
features:

  1. 1.

grain boundaries, the interface separating the grains

  2. 2.

largely zero-dimensional points called triple-junctions, wherein grain
boundaries meet.

Quadruple-junctions are another feature that accompanies grain boundaries and
triple-junctions in extremely anisotropic polycrystalline systems [23]. While
techniques to ascertain growth kinetics predominantly focus on the evolution
of grain boundaries, in the present work, a triple-junction-based attempt is
made to comprehend the rate. Accordingly, the current work comprises two
sections. In the first section, an object-detection tool is developed to
identify and count both triple- and quadruple-junctions in a polycrystalline
microstructure. This tool is subsequently used in the second section to
delineate the grain-growth kinetics.

##  2 Detecting Triple- and Quadruple-Junctions

###  2.1 Overview

---
Figure 1:  A graphical depiction of the approach adopted to develop a
regression-based tool for detecting quadruple- and triple-junctions.
Multiphase polycrystalline micrographs exhibiting grain growth rendered by
multiphase-field model is involved in training and validating the tool for
realising triple- and quadruple-junctions.

The approach employed in the present work to devise an AI tool for recognising
and counting triple- and quadruple-junctions in polycrystalline systems is
illustrated in 1. In other words, triple- and quadruple-junctions of the
polycrystalline microstructures are designated as unique objects, and are
subsequently realised by an appropriately equipped object-detection tool. A
known object-detection algorithm referred to as You-Only-Look-Once (YOLOv5)
plays a central role in realising the desired features of the polycrystalline
microstructure.

As opposed to other object-detection algorithms, YOLO operates on a
regression-based framework [24]. Simply stated, for a given polycrystalline
microstructure, a prediction is made by YOLO on the type and position of the
junction. The accuracy of this stochastic prediction is gradually enhanced,
across several iterations, by minimising the deviation from the labelled data
[25]. Repeating this process of prediction and its subsequent refinement over
different sets of microstructures essentially trains the algorithm to
accurately realise the junctions in unforeseen micrographs. The efficiency of
this training, which dictates the performance of the tool, depends on the size
of the dataset solely available for training. This need for a corpus of data
is catered, in the present work, by wide-range of numerically generated
polycrystalline microstructures. A tessellation scheme, which has been proven
to introduce physically-relevant configuration of grains, is adopted to
develop the polycrystalline microstructures [26]. These microstructures are
allowed to evolve, within the framework of multicomponent multiphase-field
model [27], thereby rendering sequence of micrographs exhibiting grain growth
and further increasing the size of the dataset. (The multicomponent
multiphase-field adopted in the present work is briefly discussed in Appendix
1 Appendix 1 - Multicomponent multiphase-field model.) Correspondingly, the
dataset includes $100$ micrographs, for every original numerically-generated
polycrystalline microstructure, capturing the different instances of the
simulated grain growth. For the ease of handling, both original and evolved
microstructures are sliced into four sections as shown in 1. Since
microstructures are devised to include atleast few hundred grains at any given
instant, even the sliced micrographs possess sufficient information for
training the algorithm. The dataset encompassing sliced original- and evolved-
microstructures is split into training and validation set in the proportion
80% and 20%, respectively. The object detection algorithm is trained to
identify and count the triple- and quadruple-junctions from the predetermined
larger portion of the dataset, while validation set is exploited to fine-tune
the accuracy through the hyperparameters.

###  2.2 Principal Dataset

---
Figure 2:  The principal dataset for the present investigation includes
multiphase polycrystalline microstructures of various phase-fractions. The
numerically-generated microstructures are turned to multiphase systems with
varying phase-fractions, and are allowed to evolve, within the framework of
multiphase-field model. The fundamental polycrystalline microstructure, along
with its temporal evolution, constitute the dataset for the present
investigation. These microstructures are designated primarily based on the
volume-fraction of the minor-phases. A set of anisotropic duplex-
microstructures are devised to introduce stable quadruple junctions.

A statistically rigorous proof of the present junction-based approach in
ascertaining growth rate is achieved by testing it against microstructures
exhibiting varied kinetics. Correspondingly, a dataset is devised which
primarily encompasses varied microstructures rendering characteristic grain-
growth kinetics. In these microstructures, a perceivable difference in the
growth kinetics is achieved by considering multiphase systems, as opposed to
homogeneous single-phase microstructure. Grains constituting the
microstructures are associated with one of two or three phases thereby
establishing polycrystalline duplex or triplex system. A characteristic phase-
fraction is introduced in a microstructure by altering the number of grains
pertaining with each phase.

Fig. 2 shows all duplex and triplex systems with unique and noticeably
distinct phase-fractions considered in the present study. Since the noticeable
difference in the grain growth kinetics stems primarily from the varying
phase-fractions, isotropic grain-boundary energy is assumed across the various
duplex and triplex microstructures. In other words, for most polycrystalline
microstructures in Fig. 2, identical and representative grain boundary energy
is assigned to the interfaces separating grains of both similar and dissimilar
phases. Interfacial energy and other parameters characterising the duplex and
triplex microstructures are tabulated in Tables 2, 3 and 4 included Appendix 1
Appendix 1 - Multicomponent multiphase-field model.

Under isotropic energy conditions, triple junctions are the only other stable
feature in polycrystalline microstructures, besides grain boundaries. Though
quadruple junctions may form during grain growth as a result of the
disappearance of grains, these features eventually dissociate to
energetically-stable triple junctions. In order to explicitly introduce stable
quadruple junctions, specific anisotropies in interfacial energies are
introduced in duplex systems with equifraction of phases, as shown in Fig. 2.
In these two-phase microstructures, anisotropy is achieved by assigning
different grain-boundary energies to the interface separating grains of like
phases, when compared to the dissimilar grains [28]. Put simply, while the
representative value of the energy is left unaltered for the interface between
the dissimilar grains, a specific value is associated with the boundaries of
same-phase grains. Based on these two types of interfacial energies, the
overall anisotropy of the microstructure is quantified as
$E_{\alpha}=\frac{\gamma_{\alpha\alpha}}{\gamma_{\alpha\beta}}$ and
$E_{\beta}=\frac{\gamma_{\beta\beta}}{\gamma_{\alpha\beta}}$. Given the focus
to introduce stable quadruple-junctions, in the present study
$E_{\alpha}=E_{\beta}(=E)$ is assumed.

As shown in Fig. 2, five different anisotropic conditions are considered by
varying energies of interface separating similar grains. As the value of
grain-boundary-energy ratio (E) increases, the number of quadruple junctions
proportionately increases [28]. Similar to other duplex and triplex
microstructures, these anisotropic systems are allowed to evolve, within the
framework of multicomponent multiphase-field model, solely to increase the
wealth of data with stable quadruple junctions.

###  2.3 Training and Validation

---
Figure 3:  Training and validation of the object-detection technique in
identifying triple- and quadruple junctions as different class of objects in
polycrystalline microstructures. The progressive improvement in the
performance of the algorithm in the present framework is assessed through loss
functions and mean average precision (MAP).

The tool to identify and count triple- and quadruple-junctions in
polycrystalline microstructures is developed by training and validation of the
object-detection algorithm. The translation of a generic detection-algorithm
to a junctions-realising tool, during training and validation, is grasped
through suitable metrics and illustrated in Fig. 3. This illustration also
includes a parameter indicating the overall performance of the junction-
detection tool on unforeseen microstructures. Epoch in Fig. 3 is related to
the iterations through batch size, which is considered to be $4$ in the
present work.

Detecting a junction in a polycrystalline microstructure begins with a
characteristic prediction made by the underlying algorithm. This prediction
assumes a form of the bounding box which supposedly encompassed the junction.
The predicted bounding box is encoded as a vector with entries representing
the coordinates of its position, dimension and probability of locating a
junction within its encapsulation. The mismatch between the prediction and
ground truth, gathered through manual labelling, is quantified as
localization- and confidence-loss. While the localization loss describes the
deviation in features of the predicted bounding-box and manual label, the
probability of an anticipated detection in comparison to the certainty of the
ground truth is expressed as confidence loss. The localization and confidence
loss cumulatively renders the box loss illustrated in Fig. 3. Exposing the
algorithm to sufficient number of polycrystalline microstructures with
manually-labelled triple and quadruple junctions, over several epochs, trains
it to detect these microstructural features. Efficient training of the
algorithm is indicated by the noticeable decrease in the box loss with epochs.
The performance of this trained junction-detecting tool on unforeseen and
relatively smaller dataset is analysed during validation. After sufficient
epochs, the accuracy of the approach in recognising triple- and quadruple-
junctions across the unknown polycrystalline structures noticeably improves.

While box loss focuses on accurately locating and encasing the junction, the
distinction between the types of junction is achieved through the class loss.
The confidence score which reflected the probability of locating any junction
within the predicted box is extended, and individual scores are included to
account for triple- and quadruple-junctions separately. These separate
confidence scores indicate the probability of finding a triple- or quadruple-
junction with a bounding box. In other words, the algorithm through different
confidence scores are trained to distinguish triple junction from the
quadruple points. Any inaccuracy in detecting the type of the junctions, in
view of the ground truth, is quantified as class loss. Similar to the box
loss, the mismatch in prediction and ground truth pertaining to the type of
junctions is weeded out through sufficient training across several epochs. The
ability to distinguish triple from quadruple junction in previously unknown
polycrystalline microstructure increases with epochs, as indicated by the
decrease in class loss in Fig. 3. During the training of the algorithm, the
hyperparameters are optimised. Correspondingly, the values 0.01, 0.0005 and
0.937 are assigned to learning rate, weight decay and momentum, respectively
[29, 30].

---
Figure 4:  A direct side-by-side comparison of the ground-truth and detection
of the trained algorithm on duplex microstructures. The ground truth on the
right of each window is established through manual labelling. Triple- and
quadruple-junctions of the anisotropic duplex microstructures are distinctly
labelled, and this distinction is accurately captured by the trained
technique.

In addition to comprehending training and validation, the overall performance
of the junction-realising tool over unforeseen microstructures is studied
through suitable parameters. The fundamental parameters adopted to unravel the
overall accuracy of the junction-tool are Recall and Precision. By considering
the ratio of true positives to the sum of true and false positives, precision
relates the accurate detection of the junctions to the inaccurate realisation.
Recall, on the other hand, indicates the accurate recognition in view of the
failed detection by normalising true positives with the some of true positives
and false negatives. Considering that the formulation of recall includes
failed detection, and precision encompasses incorrect realisation, the
optimisation of these parameters, enhances the overall performance of the
technique. This optimisation, which improves the accuracy of the detections,
is expressed as a precision-recall curve. The trajectory of the precision-
recall curve varies with the class of objects. Correspondingly, the
performance of the junction-tool is comprehensively gathered by two precision-
recall curves separately indicating the accurate identification of triple- and
quadruple-junctions. The area encapsulated by the precision-recall curve, for
a given junction (object), is referred to as the mean average precision (MAP).
The improvement in mean average precision of the tool in realising triple- and
quadruple-junctions is separately shown in Fig. 3. Given the description of
mean average precision, MAP=1.0 is the maximum value it can assume. An
absolute MAP (=1.0) value indicates a perfect performance of the junction-
realising tool characterised no inaccurate or missed detection. The present
approach through the optimisation of precision and recall offers mean average
precision of 0.897 and 0.947 for triple- and quadruple-junctions,
respectively. Considering the proximity of the observed value to the perfect
score, the performance of the developed technique is deemed convincing and is
adopted for analysing the microstructural changes.

A more straightforward representation of the performance exhibited by the
present approach in identifying junctions in polycrystalline microstructures
is offered in Fig. 4. This illustration compares the manual labelling, which
establishes the ground truth, to the detections of the developed tool. Duplex
system with grain-boundary anisotropy is specifically chosen for this
representation, given the presence of both triple- and quadruple-junctions in
its microstructure. It is evident from Fig. 4 that the algorithm is adequately
trained, and is capable of distinguishing triple- and quadruple-junctions in a
microstructure, besides collectively recognising them in an intricate
polycrystalline microstructure.

##  3 Ascertaining grain-growth kinetics

###  3.1 Inverse triple-junction plots

The junction-detection tool developed in the previous section is adopted to
analyse series of microstructures indicating temporal evolution of grains in
multiphase polycrystalline systems. Grain growth exhibited by all duplex and
triple microstructures included in Fig. 2, except anisotropic systems with
stable quadruple-junctions, are considered for this investigation. Though the
tool is capable of handling the quadruple junctions, systems with grain-
boundary anisotropy are rather overlooked in the present analysis for two
reasons. One, considering varied phase-fractions in isotropic system, a
noticeable disparity in growth kinetics is inherently expected. Secondly, a
comparison of grain-growth rate across the different microstructures is
manageable and consistent when the anisotropic systems are excluded. Moreover,
observing measurable amount of stable quadruple junctions in physical system
is extremely rare.

---
Figure 5:  Reciprocal of the number of triple-junctions, inverse count,
[T]${}^{-}1$, is plotted against time for all isotropic duplex and triplex
microstructures on the left window. Given the perceivable trend, a linear fit
is introduced in all of these plots. The statistical rigor of the inverse
count plot is enhanced by appending confidence and prediction intervals,
respectively depicted in blue and green. On the right windows, different types
of topological events are represented. Evolution of the junctions during these
topological events are highlighted.

Grain growth exhibited by 17 multiphase systems - 5 duplex and 12 triplex - is
analysed by monitoring the variation in the number of triple junctions in the
series of micrographs capturing the associated progressive change. Inverse
triple-junction count, [T]${}^{-}1$, which is the reciprocal of the number of
junction, is ascertained through the junction-detection tool for different
multiphase polycrystalline systems. The progressive change in the inverse
triple-junction count with time (unitless computational timestep [31]) for all
the duplex and triplex systems is presented in Fig. 5. A linear fit, based on
the datapoints, is introduced in all these plots. Disparity between datapoint
and the linear fit is rather marginal as discussed in Appendix Appendix 2 -
Linear fitting of inverse triple-junction datapoints using root-mean-square-
error (RMSE). In order to enhance the statistical potency of inverse count vs
time plot, confidence and prediction intervals are overlaid. These intervals
are constructed by appending additional datapoints around the existing ones.
In other words, additional datapoints, assumed to be rendered by
microstructures with similar phase-fraction but different initial
configuration, are considered around the original points. These assumed points
are normally distributed with the mean and standard deviation dictated by the
original data. The confidence interval is the mean of the assumed datapoints
surrounding the observed ones, whereas the prediction interval encapsulates
the additional datapoints themselves. The trend exhibited by the confidence
and prediction interval indicates linear relation between inverse count and
time would persist for all microstructures, even upon repeating the
simulations with different initial configurations. Put simply, both confidence
and prediction interval further substantiate the linear change in the inverse
triple-junction count with time.

Based on the prevailing trend in the temporal change of inverse triple-
junction count, across the different systems in Fig. 5, the grain-growth
kinetics can be expressed as

$[T]^{-1}(t)-[T]_{0}^{-1}=k_{T}t.$ (1)

In above relation, reciprocal of number of triple-junctions at the initial
state, and a given instant t, is denoted by $[T]_{0}^{-1}$ and $[T]^{-1}(t)$,
respectively. The characteristic kinetic coefficient, $k_{T}$ in Eqn,
quantifies the rate of grain growth in various polycrystalline system in the
present triple-junction based approach. This coefficient is the slope of
linear fit introduced in the respective plots of Fig. 5 to highlight the
apparent relation between the number of triple junctions and time during grain
growth.

###  3.2 Triple-junctions count and grain disappearance

Besides the apparent increase in the size of the grains, evolution of
polycrystalline during grain growth is accompanied by topological events.
Stated otherwise, grain growth is characterised not only by the change in the
size of the grains but also their number of sides. The changes altering the
number of sides, or the face class, of the grains are referred as topological
events. In polycrystalline microstructures, exhibiting grain growth, three
types of topological events are realised. These events are illustrated on the
right panel of Fig. 5. Topological events type-II and -III indicate sequence
of evolution wherein two grains, A and C, lose a side. This act leads to the
formation of a quadruple junction, which is energetically-unstable in
isotropic systems. Consequently, the quadruple dissociates into two triple
junctions, enabling grains B and D increase its number of sides. In other
words, type-II and III events essentially capture the loss of a side by two
abutting grains, which is eventually gained by their neighbours. Type-I event,
on the other hand, represents the topological changes associated with the
disappearance of a grain. This event, as illustrated in Fig. 5, reduces the
number of sides of top and bottom grains. When viewed topologically, triple
junctions are the sites where the sides of the grains meet. Therefore, any
change in the topological features of the grains ultimately effects the number
of triple junctions. Considering type-II and -III topological events, it is
event from Fig. 5, that despite the change in the number of sides to the
grains, the total number of triple junctions remain conserved. On the other
hand, the number of triple junctions reduces from four to two in topological
event-I. Put simply, topological events are associated either with change in
the number of sides or disappearance of grains. Events establishing change in
the number of sides largely preserve the count of triple junctions. However,
topological events accompanying disappearance of grains bring about a
noticeable change in number of triple junctions.

Though grain growth is conventionally perceived as a progressive increase in
the average grain-size, this change becomes noticeable only with the
disappearance of grains. Stated otherwise, the characterising increase in the
average grain-size is the reflection of incremental disappearance of grains.
Since the disappearance of grains, which principally quantifies grain growth,
is accompanied by change in the number of triple junctions, the inverse
triple-junction count, estimated in the present work (Fig. 5), offers a
reliable measure of the growth kinetics.

###  3.3 In view of conventional estimation

---
Figure 6:  Kinetic coefficients ascertained from the slope of the inverse
count vs time plot in Fig. 5 is presented across corresponding coefficients
calculated from the temporal change in average radius. These plots include
kinetic coefficients of all isotropic duplex- and triplex-microstructures.

The characteristic kinetic-coefficient of the present triple-junction based
approach, estimated as the slope of the linear fit reflecting the temporal
change in inverse count $[T]^{-1}(t)$, is shown in Fig. 6. This illustration
essentially encompasses grain-growth kinetics of all isotropic polycrystalline
microstructures. Beside this representation, to its right, the kinetic
coefficients estimated through the conventional approach is included in Fig.
6. This conventional kinetic-coefficient $(k_{R})$ is estimated from the
relation, $<R(t)>^{n}-<R_{0}>^{n}=k_{R}t$, where the kinetic exponent, $n=3$,
and the average radius of the polycrystalline system at the initial ($t=0$)
and at a given state ($t$) is respectively denoted by $<R_{0}>$ and $<R(t)>$
[32]. Here it is vital to note that, while $n=2$ generally reflects the rate
of grain growth in isotropic polycrystalline microstructures, in the present
systems wherein the evolution is dictated by long-range diffusion, the
exponent ($n$) assumes the value of 3 [27].  Given the grain growth in these
various systems have been numerically modelled, the average radius is
estimated from the phase-field based pixel data [33]. Both representation of
kinetic coefficients in Fig. 6, irrespective of the underlying approach,
informs the relative growth-rate exhibited by a given polycrystalline
microstructure. The noticeable similarity in the positions of the multiphase
polycrystalline microstructures across the two plot in Fig. 6 indicates the
overall agreement between the conventional and triple-junction based
estimation of growth rate. It is vital to note that, given the fundamental
difference in estimating the kinetics and the factors associated with it, an
identical spacing between the points cannot be expect across the two plots.
However, similar distribution of the points, in Fig. 6, explicates the ability
of the triple-junction based approach to render a meaningful description of
the kinetics analogous to the conventional techniques.

---
Figure 7:  Progressive change in the inverse triple-junction count ([T]-1) is
plotted against the corresponding evolution of the average radius ($<R>^{3}$).
Each set of points pertaining to a multiphase microstructure is fitted through
a line indicative of the characteristic relation between [T]-1 and $<R>^{3}$.
In the subplots, the change in radius with triple-junction counts, along with
a linear fit, is individually presented for all multiphase polycrystalline
microstructures.

Though the distribution of kinetic coefficients of different systems across
the plots are largely similar in Fig. 6, a noticeable disparity is observed in
one point. This outlying point pertains to a triplex microstructure, s12 in
Fig. 6, with predominant matrix-fraction (0.84) and scarce minor phases
(0.8$\alpha$ and 0.8$\beta$). Owing to the significant difference in the
volume fraction of the phases, grains associated with the minor phases assume
distinct positions and configuration similar to pinning-precipitates rather
than constituent grains. Consequently, the topological evolution of the grains
are seemingly affected, which in-turn introduces a deviation in the triple-
junction based estimation of growth kinetics. In other words, though triple-
junction based estimation of growth kinetics can be extended to wide range of
systems, its outcome in multiphase microstructure with exceedingly low
fraction of minor phases is rather not accurate.

###  3.4 Junctions and grain size

---
Figure 8:  Coefficient relating the inverse triple-junction counts ([T]-1) to
the average radius ($<R>^{3}$), estimated from the corresponding linear fit,
for various duplex and triplex microstructures.

An exclusive focus on junctions to estimate the growth kinetics means the
present approach rather fails to offer reasonable insight on the average grain
size of the polycrystalline microstructure. This limitation apparently
reflecting a rigid framework wherein the outcomes of the current technique
cannot be converted (or related) to the kinetics ascertained from the
conventional average-radius based approaches. In order to reduce the rigidity
of the present treatment, and enhance its versatility, the change in the
inverse junction count is analysed along with the progressive increase in
average radius ($<R>^{3}$). Stated otherwise, the progressive evolution of the
average radius ($<R>^{3}$) and triple-junction counts ([T]-1), accompanying
the grain growth, is cumulatively studied and presented in Fig. 7. This
investigation unravels a linear relation between the junction count ([T]-1)
and average radius ($<R>^{3}$) for all microstructures, irrespective of the
number of constituent phases and their fractions, which can be expressed as

$<R(t)>^{3}=\chi[T]^{-1}(t)+\text{A}([T]_{0}^{-1}).$ (2)

The coefficient $\chi$, in the above expression, relates the temporally
varying radius to the junction count. This parameter ($\chi$) is ascertained
from the linear fit of the junction count ([T]-1) and average radius
($<R>^{3}$) for all multiphase polycrystalline microstructures and is
presented in Fig. 8. Though the linear trend is independent of the phase
fraction, the coefficient assumes noticeably different values across the
duplex and triplex microstructures. While in triplex microstructures the
coefficient is around $8.5\times 10^{7}\mu m^{3}$, it increases to
$\chi=9.7\times 10^{7}\mu m^{3}$ in two-phase systems. This disparity in the
value of the relating coefficient can be attributed to the significantly
different growth kinetics exhibited by the duplex and triplex microstructures.
Besides enabling the conversion of the junction-based outcomes, the cumulative
analysis of the triple junctions and the average radius accounts for the huge
difference (order of $10^{8}$) in the kinetic coefficient of the present and
the conventional approach.

###  3.5 Bounds of the junction-based treatment

The present junction-based approach of ascertaining kinetics offers an
alternate route for both considering and correspondingly, quantifying the
evolution of polycrystalline microstructures during grain growth. This
approach can directly be adopted to gain a comprehensive understanding on the
relative differences in the grain-growth rate of a various microstructures, as
shown in Fig. 6. Furthermore, by extending the current measuring technique,
real-time change in the number of triple (or quadruple) junctions during in-
situ studies can be accurately determined. Such extension, besides obviating
the need for any dedicated apparatus, would track the growth-kinetics all
through the evolution across the entire sample. The exclusive focus on the
junctions, despite its benefits, rather implicitly precludes the present
approach from offering insights on a key parameter associated with
conventional treatments. In other words, insights on the rate of grain growth
conventionally stem from the area (size) of the grains, and subsequently,
unravel the distribution of the size in a polycrystalline system. Present
approach, reflecting its principal consideration of junctions, conveys no real
information on the grain size distribution.

By relating the growth kinetics to a quantity that is restricted to the two-
dimensional representation of the polycrystalline system, the present approach
limits itself to the analysis of 2D microstructures. Stated otherwise, the
kinetics of grain growth in the current approach is ascertained by detecting
and counting triple junctions which appear as a zero-dimensional object in the
two-dimensional microstructures. On the other hand, in three-dimension, these
features are one-dimensional entities wherein the grain-boundary surfaces
meet. Consequently, the present junction-based approach cannot be directly
extended to three-dimensional microstructures to determine growth kinetics.
However, when the intent is to understand the relative differences in the
growth kinetics across various microstructures, the current technique offers
accurate results more efficiently from the two-dimensional micrographs (Fig.
6).

##  4 Conclusion

Estimating grain-growth kinetics, which captures the rate of change in the
size of the constituent grains, is pivotal in polycrystalline systems. This
importance is primarily because the behaviour of polycrystalline materials is
noticeably influenced by the average size of the grains. Techniques generally
adopted to estimate grain-growth kinetics involves measuring average grain-
sizes of the micrographs reflecting temporal evolution of the polycrystalline
systems. Such techniques are straightforward, easy to comprehend, and offer
reliable outcome. On the other hand, besides being rather arduous, the
conventional approaches generally offer a localised estimation of the size and
consequently, demand several iterations. Even though the various steps of the
conventional techniques can be (and are) automated, the fundamental
limitations emerging from the localised focus continue to exist. Despite the
increasing role played by artificial intelligence in deciphering intricate
microstructures, these advancements are yet to shift the paradigm of grain-
growth measurement. In the present work, an approach based on object-detection
is shown to offer an alternate for estimating growth kinetics in
polycrystalline systems.

The progressive increase in the average grain-size of the polycrystalline
material during grain growth is primarily due to incremental disappearance of
grains. In other words, tracking the gradual decrease in the number of grains
would render a convincing estimation of the growth kinetics. However, such
tracking would involve monitoring the evolution of numerous grains of varying
morphology simultaneously. The change in the number of grains, besides
altering the average grain-size, change the microstructural features,
particularly the triple junctions. Put simply, the disappearance of the
grains, which dictate the increase in the average grain-size, thereby
dictating growth kinetics, is accompanied by decrease in triple junctions.
Correspondingly, an object-detection algorithm is extended to realise and
count the temporal change in the number of triple junction to render the
growth kinetics. As opposed to the conventional technique, this triple-
junction based approach considers a polycrystalline microstructure
holistically rather than locally. Moreover, inadequate etching which poses a
serious impediment to application of conventional techniques, can be handled
by the triple-junction based approach. In other words, by offering a growth
rate based on the temporal change in the number of distinguishable triple-
junctions, the effect of the inadequately etched features are minimised in the
present approach. The proven ability of the underlying algorithm to operate in
real-time suggests the potential applicability of the current technique for
the in-situ measurement of grain-growth kinetics [34].

Ascertaining the growth kinetics from the temporal change in the number
density of junctions lends itself to a unique framework in which the present
approach operates. This framework, as discussed in Sec.3.5, seemingly
overlooks certain features, particularly grain size (area) and distribution,
that constitute the foundation of conventional techniques. The lack of
explicit inclusion of grain size and distribution in the framework of
junction-based approach precludes a direct conversion growth rate determined
by one technique to another. In order to address this gap, and enhance the
versatility of the present treatment, attempts will be made in the subsequent
works to first establish a relation between the junction count and size of the
individual grains. A deeper understanding of this relation between the
topological and geometrical feature would facilitate in the cross-verification
of the growth kinetics estimated from different techniques.

## Appendices

### Appendix 1 - Multicomponent multiphase-field model

In multiphase-field modelling of grain growth, each grain is distinguished by
a scalar variable called phase-field such the evolution is realised by the
spatio-temporal change in this variable not by the interfaces. Since the
multiphase system is involved in this work, the conventional description of
polycrystalline microstructure is extended and distinct phases are associated
with the grains. Correspondingly, the phase-field variables representing the
grains in a multiphase set-up adopt a form

$\displaystyle\mbox{{\bm{$\phi$}}}=\Big{\\{}\underbrace{\\{\phi_{\alpha}^{1},\phi_{\alpha}^{2}\dots\phi_{\alpha}^{q_{\alpha}}\\}}_{\bm{\phi_{\alpha}}},\underbrace{\\{\phi_{\beta}^{1},\phi_{\beta}^{2}\dots\phi_{\beta}^{q_{\beta}}\\}}_{\bm{\phi_{\beta}}}\dots\underbrace{\\{\phi_{N}^{1},\phi_{N}^{2}\dots\phi_{N}^{q_{N}}\\}}_{\bm{\phi_{N}}}\Big{\\}}.$
(3)

Associating phase to a grain involves assigning a characteristic composition,
$\\{\mbox{{\bm{$\phi$}}}(\mbox{{\bm{$c$}}})=\mbox{{\bm{$\phi$}}}_{\alpha}(\mbox{{\bm{$c$}}})|\mbox{{\bm{$c$}}}=\bm{c}_{\alpha}\\}$,
where $\bm{c}_{\alpha}$ includes all alloying elements. The concentration of
the phases are specifically realised and assigned to ensure that chemical
equilibrium exists between the different phases and any evolution is primarily
governed by the inherent difference in the curvature, id est the grain growth.
Concentrations establishing equilibrium across the phases are tabulated in
Table. 2.

Despite the assuming same concentration, the grains of similar phases are
distinguished by the characteristic phase-field. These distinct phase-fields
avoid coalescence and ensure a stable grain boundary between the chemically-
similar grains.

Based on the characteristic phase-fields establishing multiphase
polycrystalline microstructure, the overall energy-density of the system 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$}}})$
(4)
$\displaystyle=\int_{V}f_{\text{int}}(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})+f_{\text{bulk}}(\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})\mathrm{d}{V},$

where
$\int_{V}f_{\text{int}}(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})$
and $f_{\text{bulk}}(\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})\mathrm{d}{V}$
represent the energy-density contribution of the diffuse interface and bulk
phase-associated grains, respectively. The spatio-temporal evolution of phase-
field which translates to the evolution of the grains is modelled by
minimising the overall energy-density of the system. Therefore, phase-field
evolution is solved by

$\tau\varepsilon\frac{\partial\phi_{\alpha}^{m}}{\partial
t}=-\frac{\partial\cal{F}(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}},\mbox{{\bm{$c$}}})}{\partial\phi_{\alpha}^{m}}=\varepsilon\left[\bm{\nabla}\cdot\frac{\partial
a(\mbox{{\bm{$\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$}}})}{\partial\phi_{\alpha}^{m}}\right]-\left[\frac{{\partial}f^{\alpha}_{m}(\mbox{{\bm{$c$}}}^{\alpha},\phi_{\alpha}^{m})}{\partial\phi_{\alpha}^{m}}\right]-\Lambda,$
(5)

where $\phi_{\alpha}^{m}$ indicates grain-$m$ associated with phase-$\alpha$.
Terms $w(\mbox{{\bm{$\phi$}}})$ and
$a(\mbox{{\bm{$\phi$}}},\bm{\nabla}\mbox{{\bm{$\phi$}}})$ denote penalising
potential and gradient energy with
$f^{\alpha}_{m}(\mbox{{\bm{$c$}}}^{\alpha},\phi_{\alpha}^{m})$ indicating
equilibrium free-energy of grain-$m$. At any given point, sum of all the
constituent phase-fields should be one. This characteristic criterion of the
multiphase-field approach is ensured by the Lagrange multiplier, $\Lambda$.

Table 2: Concentration of phases establishing chemically equilibrium system Phase | Independent component-$i$ | Independent component-$j$
---|---|---
(Binary) |  |
Phase-$\alpha$ | 0.1 | -
Phase-$\beta$ | 0.9 | -
(Ternary) |  |
Phase-$\alpha$ | 0.05 | 0.05
Phase-$\beta$ | 0.05 | 0.9
Phase-$\gamma$ | 0.9 | 0.05

The spatio-temporal evolution of the dynamic variable, chemical potential
($\mu_{i}$), establishing equilibrium and curvature-based migration of the
chemical components reads

$\frac{\partial\mu_{i}}{\partial
t}=\left\\{\bm{\nabla}\cdot\left[\sum_{j=1}^{K-1}\bm{M}(\mbox{{\bm{$\phi$}}})\bm{\nabla}\mu_{j}\right]-\sum_{\alpha}^{N}\sum_{m}^{q}c_{i}^{\alpha}\frac{\partial\phi_{\alpha}^{m}}{\partial
t}\right\\}\left[\sum_{\alpha}^{N}\sum_{m}^{q}h(\phi_{\alpha}^{m})\frac{\partial
c_{m:i}^{\alpha}}{\partial\mu_{j}}\right]_{ij}^{-1},$ (6)

where $\bm{M}(\mbox{{\bm{$\phi$}}})$ is matrix encompassing mobility of the
migration chemical species. Table. 3 lists the constants assumed to model all
multiphase polycrystalline systems.

Table 3: Material parameters adopted to model duplex and triplex microstructures Parameter | Symbol | Value
---|---|---
Grain boundary energy | $\bar{\gamma}_{\alpha\alpha}=\bar{\gamma}_{\beta\beta}\bar{\gamma}_{\gamma\gamma}$ | 1.0 $\text{Jm}^{2}$
Interphase boundary energy | $\bar{\gamma}_{\alpha\beta}=\bar{\gamma}_{\beta\gamma}=\bar{\gamma}_{\alpha\gamma}$ | 1.0 $\text{Jm}^{2}$
Bulk diffusivities | $D^{\alpha}=D^{\beta}=D^{\gamma}$ | 1.0 $m^{2}s^{-1}$

Despite the varying phase-fractions, all multiphase polycrystalline
microstructures are intialised and their evolutions are modelled in two-
dimensional domains of identical configurations Through finite-difference
scheme, the two-dimensional domains are discretised into $2048\times 2048$
grids of dimensions, $\Delta$x = $\Delta$y = $5\times 10^{-7}$m uniformly.
Polycrystalline setup over these discretised grids are established by Voronoi
tessellation rendering approximately $10000$ grains.

Table 4: Parameters associated with the multiphase-field simulations Parameter | Symbol | Value
---|---|---
Timestep width (No unit [27]) |  $\Delta$t | 1.0
Interface-width pre-factor | $\varepsilon$ | 0.2 $\leavevmode\nobreak\ \mu\text{m}$ [31]
Relaxation parameter | $\tau$ | 1.0 $\text{Jsm}^{-4}$
---
Figure 9:  The deviation of the datapoints from fitting introduced in all
inverse count vs time plots presented in Fig. 5 is quantified as RMSE (Root-
Mean-Squared-Error).

Microstructural evolution of the multiphase polycrystalline system is modelled
by solving temporal evolution of the dynamic variables - phase-field and
chemical potential of components - by forward-marching Euler’s scheme over the
homogeneously discretised domain. The simulation domain is decomposed into
sections of smaller dimensions, and handled simultaneously by Message Passing
Interface (MPI). The involvement MPI ensures efficient use of the
computational resources.

### Appendix 2 - Linear fitting of inverse triple-junction datapoints

Upon of introducing a linear fit to the datapoints of all microstructures in
Fig. 9, the disparity between the individual point and the fitted line is
estimated as RMSE (Root-Mean-Squared-Error). These deviations between fitting
and datapoints, quantified as RMSE, for different microstructures are
illustrated in Fig. 9. This depiction unravels the RMSE value for all the
multiphase systems fall within $1.0^{-5}$. In other words, irrespective of the
phase-fractions, minimal disparity exist between the linear fitting and
datapoints in all multiphase systems. Consequently, the linear relation is
adopted to describe the change in inverse triple-junction count with time.

## Data Availability

The source code for detecting junctions in polycrystalline microstructures and
sample images employed in the present work are made available at
https://github.com/theormets/Junction-based-Grain-Growth-Measurement

## Declaration of Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgments

PGK Amos thanks the financial support of the SCIENCE & ENGINEERING RESEARCH
BOARD (SERB) under the project SRG/2021/000092.

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# BEAM–BEAM STUDY OF ERL BASED eRHIC

Y. Hao  V.N. Litvinenko  V. Ptitsyn  BNL  Upton  NY  USA

###### Abstract

Beam–beam effects in eRHIC, the proposed ERL-based Electron–Ion Collider (EIC)
at BNL, have several unique features distinguishing them from those in hadron-
colliders and lepton-colliders. Taking the advantage of the fact that the
electron beam is used only once, we expect the luminosity to be 10 times
greater than for the ring–ring collision scheme with similar parameters.
However, without instituting proper treatments, the quality of electron and
hadron beams can undergo degradation or even beam loss, driven by the
beam–beam interactions. We will discuss the harmful effects, including the
disruption and mismatch effect of the electron beam, the kink instability and
the noise heating of the ion beam and the possible countermeasures.

##  1 Introduction

The main advantage of an energy recovery linac (ERL) based electron–ion
collider (EIC) compared with a ring–ring collider is the higher achievable
luminosity of the former. In an ERL-based EIC, which we also call a linac-ring
scheme, the electron bunch collides only once with the ion bunch and
thereafter is recycled. Hence, the beam–beam parameter for the electrons in
ERL scheme can exceed by a large margin (as in Table 1) that permissible for
electron circulating in a ring. While the beam–beam parameter for the ions
remains the same in both schemes, the luminosity achieved in the linac-ring
collision scheme exceeds that of the ring–ring collider scheme between 10 and
100 times [1]. Figure 1 illustrates the layout of eRHIC, the EIC proposed in
Brookhaven National Laboratory. Table 1 lists its design parameters.

In the new parameter range of eRHIC, the electron beam is subject to a very
strong beam–beam effects that create a new set of beam dynamics effects.
First, the electron beam experiences significant disruption and mismatch
effects due to the beam–beam interaction. Second, the collective motion of the
electron beam inside the ion beam during their collision can cause a new
head–tail type of instability, named ’kink instability’. And the ion beam can
be heated up by the noise of the fresh electron beam each turn. In this paper,
we will report our studies on those individual effects and carry out
countermeasures to the harmful ones.

Figure 1: eRHIC design layout. The blue and yellow curves represent the existing blue and yellow rings of RHIC. The red curve illustrates the new ERL accelerator for the electron beam. Table 1: Parameter range of eRHIC Parameters |  Range
---|---
Electron beam energy (GeV) |  5-30
Ion beam energy (GeV) (proton) |  50-250
Electron beam disruption parameter |  5-142
Ion beam–beam parameter |  0.015
Ion bunch length (cm) |  8.3
Electron bunch length (cm) |  0.2-0.4
Electron and ion $\beta^{*}$ (cm) |  5
Ion synchrotron tune |  0.004

##  2 Electron disruption effects

The electron beam experiences very strong beam–beam force from the ion beam in
the interaction region. The force will make the electron beam oscillate inside
the ion beam and deform the distribution of the electron beam, as studied in
[2]. We found that the disruption parameter $d_{e}=l_{i,z}/f_{e}$ is
convenient to characterize the oscillation of the electron beam, where
$l_{i,z}$ is the ion bunch length and $f_{e}$ is the focal length of the
linearized beam–beam interaction. For an ion beam with Gaussian longitudinal
distribution, the number of oscillations $n$ of the electron beam inside the
ion beam is

$n=\frac{\sqrt{d_{e}}}{\left(2\pi\right)^{3/4}}\approx\frac{\sqrt{d_{e}}}{4}.$

Thus, for the eRHIC parameters, a single electron will oscillates up to 3 full
oscillations in the ion beam.

Figure 2: The top figure shows the phase space distribution of the electron
beam after collision, and bottom figure shows the evolution of the electron
beam size and emittance, for $d_{e}=27$. In the top figure, the r.m.s. and 6
r.m.s. ellipses for both geometric and effective emittance, respectively, are
plotted.

Figure 3: The top figure shows the phase space distribution of the electron
beam after collision, and bottom figure shows the evolution of the electron
beam size and emittance, for $d_{e}=150$. In the top figure, the r.m.s. and
6r.m.s. ellipses for both geometric and effective emittance, respectively, are
plotted.

We use simulation code, EPIC [3], to calculate the electron beam evolution
inside the opposing ion beam. Figure 2 and Fig. 3 illustrate the examples of
the electron beam distribution after the collision and the e-beam evolution
inside the ion beam. The former correspond to the case of $d_{e}=27$, and
latter for $d_{e}=150$. In the electron beam distribution plots, the nonlinear
force deform its initial Gaussian distribution completely. The electrons with
larger betatron amplitude rotate slower than those in the core. Therefore the
distribution after collision forms a spiral shape. We use 2 different
definitions of beam emittance to characterize the occupied phase space area.
One is the r.m.s. geometric emittance obtained from the beam distribution,
written as

$\varepsilon_{x}=\sqrt{\left\langle{\left({x-\bar{x}}\right)^{2}}\right\rangle\left\langle{\left({x^{\prime}-\bar{x}^{\prime}}\right)^{2}}\right\rangle-\left\langle{\left({x-\bar{x}}\right)\left({x^{\prime}-\bar{x}^{\prime}}\right)}\right\rangle^{2}}.$
(1)

The other emittance uses the design optics function and is called effective
emittance. It is defined as the half of the average value of the
Courant–Snyder invariant of all macro-particles based on the design lattice

$C\left({\tilde{x},\tilde{x}^{\prime}}\right)=\gamma\tilde{x}^{2}+2\alpha\tilde{x}\tilde{x}^{\prime}+\beta\tilde{x}^{\prime
2}.$ (2)

In the e-beam distribution plots of Fig. 2 and Fig. 3, both emittances are
represented as ellipses of 1 r.m.s. value and 6 r.m.s. value. The evolution
plots illustrate the evolution of the 2 r.m.s. emittance and the r.m.s. beam
size. These plots clearly show the mismatch between the beam distribution and
the design optics due to the beam–beam interaction. The effective emittance
will determine the aperture requirement of the magnet downstream of
interaction point (IP), as shown in Fig. 4. The calculated aperture shows that
the small-gap magnet designed for eRHIC is suitable for the ERL energy
recovery passes.

Figure 4: The aperture requirement is shown of the energy recovery pass
downstream IP. A maximum 10 m $\beta^{*}$ is assumed in all arcs.

##  3 Kink instability and its mitigation methods

The kink instability develops due to the electron beam passes the imperfection
of the head of the ion beam to its tail. Therefore, for the ion beam, the
beam–beam interaction behaves as an effective wake field. If we assume both
beams have only infinitesimal offsets, the wake field

$W\left(s,s^{\prime}\right)=\dfrac{\gamma_{i}}{Z^{2}N_{ib}r_{i}}\dfrac{\Delta
x^{\prime}\left(s\right)}{\Delta x\left(s^{\prime}\right)}$ (3)

can be retrieved from simulation, where $N_{ib}$ is the number of ions in the
slice, $\gamma_{i}$ is the energy of the ion beam and $r_{i}$ is the classical
radius of the ion beam. The wake field is illustrated in Fig. 5.

Figure 5: The example of the kink wake field with the beam–beam parameter of
the ion beam $\xi_{p}=0.015$ is shown. The electron beam has disruption
parameter $d_{e}=27$. In the figure, the electron beam travels from the
positive $s$ to negative. The symbol $s^{{}^{\prime}}$ denotes the location of
the introduced offset.

The threshold of the strong head–tail instability (the kink instability) can
be calculated using the 2-particle model or the multi-particle model[4]. Both
models are based on linearized beam–beam forces. For a 2-particle model, the
threshold is simply: $\xi_{i}d_{e}<4\nu_{s}/\pi$. However, to model the
electron beam correctly in the high disruption parameter case, the multi-
particle model should be used, predicting the threshold as in Fig. 6.

Figure 6: The threshold of kink instability, with the choice of the
synchrotron tune 0.004, is shown. The blue dots denote the threshold
calculated from the 51 macro-particles circulant matrix method. The red line
represents the simple threshold form from equation
$\xi_{i}d_{e}<4\nu_{s}/\pi$. The green line corresponds to $\xi_{i}=0.015$,
which is the design beam–beam parameter of ion beam in eRHIC.

Both linear models predict that the parameter of the eRHIC exceeds the
threshold. A simulation using nonlinear beam–beam forces is required to
confirm this understanding. Figure 7 shows the emittance growth associated
with the kink instability at different disruption parameters of the electron
beam. Even with the lowest disruption parameter, $d_{e}=5$, the system is not
stable at +2 chromaticity (the nominal value of RHIC operation), although the
emittance growth in this case is much less than those with higher $d_{e}$. If
we increase the chromaticity to stabilize the emittance growth, it requires
unreasonable values. Therefore, a dedicate feedback system is desired as a
countermeasure.

Figure 7: The top figure shows the proton beam emittance growth due to the
kink instability at different disruption parameters with the chromaticity of
both transverse directions set at $+2$ units, and the beam–beam parameter of
the proton beam at 0.015. The bottom figure shows Fourier spectrum of the turn
by turn proton slice centroid data. The proton beam is cut to 100 longitudinal
slices for this calculation.

The first feedback system [4], shown in Fig. 8, takes full advantage of
flexibility of a linac-ring scheme, which has the following procedures. We
steer the fresh electron bunch before collision based on the transverse offset
of the last-used electron bunch that collides with the same ion bunch. Then
the centroid of the new electron bunch will oscillate within the opposing ion
bunch due to the focusing beam–beam force. We are expecting that oscillation
of the centroid of the electron bunch gives the ion bunch proper kicks to
correct the offset of the ion bunch before the visible adverse effect, such as
emittance growth and luminosity loss, due to the kink instability.

Figure 8: The schematic layout is shown of the feedback system I for
mitigating the kink instability in eRHIC.

Mathematically, we introduce the offset by modifying the motion of the
centroid of the electron bunch based on the information from the last one:

$\binom{\bar{x}_{e}}{\bar{x}_{e}^{\prime}}_{n+1,i}=M_{f}\binom{\bar{x}_{e}}{\bar{x}_{e}^{\prime}}_{n,f}.$
(4)

Here, the subscript $n$ denotes the electron–ion collision in $n^{th}$ turn,
and the subscripts $i$ and $f$ respectively represent the bunch centroid
before and after collision. Map $M_{f}$ defines the algorithm of the feedback
system. Here, for simplicity and easier realization, we limit $M_{f}$ to be a
matrix.

Figure 9: The effect of the feedback system at disruption parameter 5 is
shown. Figure 10: The effect of the feedback system at disruption parameter 20
is shown.

Figure 9 shows the effect of this feedback system at disruption parameter 5.
In this case, the emittance growth due to the kink instability is suppressed
with proper amplitude of the feedback gain $m_{11}$ (-0.01 or -0.02) without a
noticeable decreasing in luminosity. An incorrect sign of the gain may boost
the instability, as shown by the red dots in Fig. 9.

When the disruption parameter exceeds 15, this feedback system itself can not
stabilize the emittance, because the system will excite the instability of the
rigid mode while it can correct the head–tail mode of the ion beam. Therefore
we have to add the transverse bunch-by-bunch damper to damp the rigid mode of
the ion beam simultaneously. The result for $d_{e}=20$, as an example, is
shown in Fig. 10. The red dots show the case with the feedback gain of
$m_{11}=0.03$ without transverse damper. The centroid of the ion bunch becomes
unstable and causes fast emittance growth due to the offset of two beams. By
applying the bunch-by-bunch feedback in the simulation, the ion centroid is
stable and the emittance growth is prevented (blue curve).

The simple feedback loses its efficiency when $d_{e}>25$. In this range, the
electron beam oscillate too fast and the frequency of the oscillation does not
match that of the lowest instability mode. We need an alternative feedback
scheme for this disruption parameter range, such as a traditional pick-up and
kicker system in the ion beam, to suppress the instability coherently [5].

For the pickup–kicker system, the effect can also be modelled as a wake field.
If we assume the system has a uniform frequency response with low and high
frequency limits $f_{L}$ and $f_{H}$, the corresponding wake field of this
system is [6]

$W\left(\tau\right)=R\int_{f_{L}}^{f_{H}}\cos\left(2\pi f\tau\right)\,df,$ (5)

where $R$ is related to gain of the amplifier between the pickup and the
kicker.

We fix the low frequency limit to 50 MHz, which is below the first peak in the
bottom figure of Fig. 7. Then we vary the high frequency limit to find the
requirement for the individual disruption parameter.

Figure 11 shows that the required $f_{H}$ is at least 2.1 GHz to suppress the
kink instability when $d_{e}$ is 150. For other $d_{e}$, as shown in Fig. 12,
the required $f_{H}$ is a monotonically increasing function of $d_{e}$.
Therefore, we demonstrated that the kink instability will be suppressed by a
pickup and kicker system with whole electron beam disruption parameter range
(5–150), if the required frequency bandwidth is selected.

Figure 11: The comparison is shown of kink instability dampings with different
high frequency limits $f_{H}$ when the disruption parameter $d_{e}=150$. The
gain of the feedback is selected to minimize the emittance growth ion beam.
Figure 12: The relation is shown between the required high frequency limit
$f_{H}$ and the electron disruption parameter $d_{e}$. Each point shows where
the instability can be suppressed in the corresponding parameter ($f_{H}$ and
$d_{e}$) with proper amplitude. For all calculations, the low frequency limit
is set at 50 MHz

##  4 Noise heating effect of the ion beam

Since the ion beam always collides with fresh electron bunches, the electron
beam parameter fluctuation will affect the ion beam through the beam–beam
interaction. The fluctuations can be classified as two types. The first is
dipole errors due to the electron beam transverse position offset; the second
is quadrupole error due to the fluctuation of the electron beam intensity or
transverse beam size.

If the noise of the electron beam is considered as white noise, i.e. a uniform
spectrum in frequency domain, the effect of both dipole error and quadrupole
errors can be evaluated analytically. For the quadrupole errors, the r.m.s.
beam size of the ion beam is expected to grow exponentially, with the rising
time

$\tau=\frac{T}{4\pi^{2}\xi_{i}^{2}}\frac{1}{\left(\delta f/f\right)^{2}},$

where $\xi_{i}$ is the beam–beam parameter of the ion beam, $T$ is the
revolution period and $\delta f/f$ is the r.m.s. error of the beam–beam focal
length. For eRHIC parameters, to get a slow rising time (~10 hours), the
relative error of the electron beam parameter should be better than $2\times
10^{-4}$. A Lorentz frequency spectrum
$g(\omega)=1/(\omega^{2}+\alpha^{2}\omega_{0}^{2})$ is considered, where
$\alpha$ is a free parameter much less than 1 and $\omega_{0}$ is the angular
revolution frequency of the ion ring. In this case, the rising time $\tau$ is
lengthened to $\tau/R(\alpha)$, where

$\displaystyle R(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1-\exp(-2\alpha)}{1+\exp(-2\alpha)-2\cos(4\pi\nu)\exp(-\alpha)}$
$\displaystyle=$ $\displaystyle\frac{\alpha}{1-\cos(4\pi\nu)}+O(\alpha^{3}).$

For the dipole errors, the ion beam is kicked turn by turn due to the electron
beam random offset. By following the well-known random walk formulas, the ion
beam displacement gives $\sqrt{<x_{i}^{2}>(t)}=\sqrt{t/\tau+<x_{i}^{2}>(0)}$
and $1/\tau=8\pi^{2}\xi_{i}^{2}<d_{n}^{2}>/T$, where $d_{n}$ is the $n^{th}$
turn electron beam displacement at IP. We need a bunch-by-bunch transverse
damper in the ion ring to compensate the dipole heating up effect.

##  5 Conclusion

We report on the key finding for distinct beam–beam effects in the ERL based
eRHIC. Our study identified the challenges as well as possible countermeasures
for both the electron and the ion beams.

A dedicated feedback system is required to suppress the emittance growth
caused by the kink instability. We proposed two possible feedback systems. The
feedback applied to the electron beam works for moderate values of the
disruption parameter, e.g. $d_{e}<25$. A traditional broad-band pickup and
kicker feedback system can damp the instability for the whole range of the
disruption parameter expected in eRHIC.

We report on the requirement for the intensity and beam size stability of the
electron beam to avoid the hadron beam emittance growth caused by noise in
beam–beam interactions. We also established a need for a transverse bunch-by-
bunch damper to compensate for the possible heating effect caused by random
noise in the transverse displacement in the electron beam.

## References

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  * [2] Y. Hao and V. Ptitsyn, _Phys. Rev. ST Accel. Beams_ 13 (2010) 071003.
  * [3] Y. Hao, Beam-Beam Interaction Study in ERL based eR- HIC, Ph.D. thesis, Indiana University, 2008.
  * [4] Y. Hao, V. N. Litvinenko and V. Ptitsyn, _Phys. Rev. ST Accel. and Beams_ 16 (2013) 101001.
  * [5] Y. Hao _et al._ , Kink Instability Suppression with Stochastic Cooling Pickup and Kicker, Proc. Int. Particle Accelerator Conf., New Orleans, NM, 2012.
  * [6] M. Blaskiewicz and J.M. Brennan, WEM2105, Proc.COOL, Bad Kreuznach, 2007.