# Three-dimensional morphology of an ultrafine Al-Si eutectic produced via
laser rapid solidification

Xinyi Zhou 111These authors contributed equally: Xinyi Zhou and Paul Chao Paul
Chao 222These authors contributed equally: Xinyi Zhou and Paul Chao Luke Sloan
Huai-Hsun Lien  Allen H. Hunter  Amit Misra  and Ashwin J. Shahani
<EMAIL_ADDRESS>Engineering Physics Program, University of Michigan, Ann
Arbor, MI 48109, USA  Department of Materials Science and Engineering,
University of Michigan, Ann Arbor, MI 48109, USA  Michigan Center for
Materials Characterization, University of Michigan, Ann Arbor, MI 48109, USA
Department of Chemical Engineering, University of Michigan, Ann Arbor, MI
48109, USA

###### Abstract

Al-Si alloys processed by laser rapid solidification yield eutectic
microstructures with ultrafine and interconnected fibers. Such fibrous
structures have long been thought to bear resemblance to those formed in
impurity-doped alloys upon conventional casting. Here, we show that any
similarity is purely superficial. By harnessing high-throughput
characterization and computer vision techniques, we perform a three-
dimensional analysis of the branching behavior of the ultrafine eutectic and
compare it against an impurity-modified eutectic as well as a random fractal
(as a benchmark). Differences in the branching statistics point to different
microstructural origins of the impurity- and quench-modified eutectic. Our
quantitative approach is not limited to the data presented here but can be
used to extract abstract information from other volumetric datasets, without
customization.

###### keywords:

Rapid solidification , Laser surface remelting , 3D microstructure , eutectic

††journal: Scripta Materialia

Al-Si eutectic alloys are widely used as structural materials in automotive
and aerospace applications due to their high strength-to-weight ratio and
excellent castability [1]. Their mechanical properties such as tensile
strength and ductility can be further improved by avoiding the formation of
coarse, brittle Si flakes during solidification [2, 3, 4, 5]. This is
accomplished via impurity-modification, in which the introduction of a few
hundred ppm Na or Sr leads to a refinement of the Si into a fibrous structure
on the order of micrometers [6, 7]. On the other hand, rapid solidification
alone gives rise to a microstructural refinement [8, 9, 10, 11, 12]: for
example, laser surface remelting creates rapid cooling conditions (as high as
1010 K/s [13]) to refine the microscale flakes to nanoscale fibers [14, 15,
16, 17]. Such a quench-modification is promoted above a critical
solidification front velocity [18, 19]. In this context, the term
“modification” indicates a dramatic alteration in the morphology of Si that is
produced by eutectic solidification.

It has generally been assumed by some authors [18, 20, 21, 22, 23] that the
two routes of modification lead to the same end-product (namely, fibers of Si
in a matrix of Al). Yet this assumption has never been formally tested. Two-
dimensional (2D) images of fibrous Si (see Fig. S1 for an example) may present
similar features (e.g., aspect ratios [19]), but the morphology and topology
of the Si phase may be distinct in three-dimensions (3D). Importantly, Requena
and coworkers [24, 25] have found that the load-carrying capacity of Al-Si
alloys depends on the internal 3D architecture, i.e., the connectivity of the
eutectic Si phase. Only recently is it possible to obtain a detailed
characterization and quantitative, comparative analysis of these structures at
the nanometric level, due to advances in high resolution and high throughput
3D imaging and development of novel data-driven microstructure analysis tools.
Computer vision provides an opportunity to construct rich 3D microstructural
fingerprints [26], which can be analyzed and compared for the two mentioned
processing routes, and without significant assumption or human intervention.

Here, we leverage the above capabilities to quantify the 3D morphology of
ultra-fine Si formed by laser quench-modification [15] and compare to ($i$) Si
formed by impurity-modification in directional solidification (previously
characterized in [26]) and to ($ii$) a random fractal simulated via diffusion
limited aggregation (DLA). The laser remelted alloy was of composition
Al-20wt%Si, cast from Al (99.99%) and Si (99.99%), see Ref. [15] for
additional details. The reason we selected a hyper-eutectic alloy for our
quench experiment was to ensure that we remain within the (skewed) coupled
zone of Al-Si [14], at the relatively high velocities associated with laser
rapid solidification. In case ($i$), the directionally solidified Al-7wt%Si
alloy contains 150 ppm of Sr [26] (i.e., above the concentration threshold for
Sr modification [27]) and indeed it shows typical impurity-modified
morphological characteristics and the basic parameters listed in Table 1. Our
analysis of the branching behavior333The branched structures presented herein
resemble at first glance a dendritic morphology, but note they are much
smaller than the primary Si dendrites observed in, e.g., Ref. [28]. offers
insight into their morphological differences and formation mechanisms.

To characterize the connectivity of fine fibrous Si, we utilize focused ion
beam (FIB) tomography. Through serial-sectioning, we can reconstruct the 3D
microstructure via scanning electron microscopy (SEM) [29]. This method is
advantageous since it can accommodate volumes larger than
$1000\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{3}$ with voxels
approaching tens of nm compared to X-ray nanotomography, in which the sample
dimension is limited to $\sim$$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [29,
30]. Also, the similar x-ray attenuation coefficients of Al and Si make it
challenging to distinguish between them at ever decreasing length-scales. We
used the ThermoFisher Helios 650 Nanolab SEM/FIB (ThermoFisher Scientific,
Hillsboro, OR) for milling and imaging. The ion beam operated at an energy of
30 keV and the electron beam at an energy of 5 keV. We selected a region at
the center of the laser surface remelted (LSR) zone of the same Al-Si sample
described in Ref. [15], see Fig. 1(a). In total, we collected 512 images with
6.5 nm pixel size at 5 nm intervals along the milling direction (corresponding
to the laser direction). See Fig. 1(b) for a representative FIB cross-
sectional image and Fig. 1(c) for an orthogonal view.

Following data acquisition, we removed ion milling “curtain” artifacts in each
cross-sectional SEM image using a compressed sensing stripe removal algorithm
[31]. Fig. 1(b) is a representative image after stripe removal, showing Si
fibers with circular cross-section. Then, we stretched all images vertically
to account for the 52° sample tilt during data acquisition and registered the
images using phase correlation in Matlab [32], considering only rigid
translations. After processing the data, we utilized Zeiss Zen Blue 3.1 with
the Intellesis deep learning module [33, 34] to perform semantic segmentation,
a pixel-based machine learning method, to identify the solid phases in each
SEM cross-section. For this purpose, we supplied five manually segmented
images as training data. After training, the model was capable of
distinguishing Si from Al. Fig. 1d shows a reconstructed subvolume, depicting
the complex and bicontinuous fibrous Si (see Video. V1 for a visualization of
a larger volume).

Figure 1: LSR produces an ultra-fine Al-Si eutectic microstructure imaged by
SEM-FIB tomography: (a) melt pool boundary (demarcated by white dashed line)
and the location we selected for FIB tomography (red box). The laser moves
along the $y$ direction at a speed of 30 mm/s. (b) SEM image of a FIB cross-
section from the fibrous eutectic microstructure at the center of the melt
pool. Dark spots are the cross-sections of Si fibers while the lighter regions
correspond to the Al matrix. (c) An etched region that illustrates the coral-
like structure within the melt pool, reproduced with permission from [15]. (d)
3D reconstruction that displays the complex and bicontinuous morphology of
eutectic Si structure.

Due to the interconnectivity of the ultra-fine Si structure, it is impossible
and also unreasonable to interrogate the morphology without having a clear
starting and end point within the dataset. To make the problem tractable and
meaningful, we decide to isolate a single colony for further analysis (as
opposed to the aggregate network). These structures form as a result of growth
front instability when the ratio of the velocity $V$ to the thermal gradient
$G$ exceeds a critical value, thus destabilizing the otherwise planar front
[35, 36]. They are common in laser treated materials since the generally high
$V/G$ ratio establishes proper conditions for the formation of cells, even in
high-purity alloys, see, e.g., [16].444While we have not conducted a detailed
chemical analysis, we can estimate the concentration of impurities in our
alloy using the constitutional supercooling criterion [37]: assuming $V=30$
mm/s (from Table 2) and $G=300$ K/mm (order-of-magnitude, from Ref. [38]), and
taking the thermophysical parameters from Ref. [39], we would expect $\sim$10
ppm Sr is enough to destabilize the Al-Si eutectic growth front and generate
cells via constitutional supercooling. That said, this low concentration is
well below that needed for impurity modification, which gives rise to
refinement and twinning (for an Al20wt%Si alloy, the critical concentration of
Sr for modification is greater than 100 ppm, according to thermodynamic
calculations in Ref. [27]). The cells or colonies have a coarser eutectic
microstructure at their boundaries compared to their interior [35].
Unfortunately, however, attempts at manually outlining the colonies were
tedious and subject to bias. To overcome this challenge, and inspired by Ref.
[40], we quantify several microstructural descriptors of each connected
component of Si (e.g., its volume, surface area, and aspect ratio) via Matlab
[32]. Then, we determine the contribution of each descriptor (after
normalization) to the overall variance, via principal component analysis. This
enables us to identify those features that distinguish the Si fibers at the
boundary from all others and ultimately to trace the colony boundaries in 3D,
see supplementary information, Fig. S2. A selected colony is highlighted in
red in Fig. 2(a) and visualized in 3D in Fig. 2(b).

Figure 2: Hierarchical eutectic microstructure: (a) a 2D slice of the FIB
tomography dataset with lines (in red) that correspond to the boundaries of
the colonies. We use a machine learning procedure to identify the lines (see
text for details). (b) Single colony that we extract from the 3D data, the
same as that shaded in (a). (c) Single tree-like structure within (b) that is
branching out isotropically in 3D, from bottom-to-top. (d) Skeleton of the
tree structure in (c). The tree can be decomposed into nodes (blue) connected
by links (red). The edge length is defined as the length of each link between
adjacent nodes and the stem length is the distance from the starting “seed” of
a single tree (at bottom) to all the end-points of the branches.

Subsequently, we skeletonize the single colony containing ultra-fine Si fibers
via medial axis thinning [41], in order to quantify the branching behavior.
The decomposed network structure can be represented by a list of nodes (with a
defined spatial location) and links (spanning two nodes). Despite this reduced
representation, we obtain a “forest” of nodes and links for the highly
interconnected Si fibers within the colony. To make our task more manageable,
we subdivide the Si fibers inside the colony into individual “trees” following
two empirical rules (see Fig. S3 and Video. V1). Each tree has characteristic
edge lengths and a stem length (Fig. 2(c,d)), defined in the same way as in
Ref. [26].

We are now positioned to quantitatively compare the quench-modified eutectic
against the impurity-modified eutectic (from Ref. [26]), with respect to
length-scale and morphology.

Fig. 3(a,b) shows distributions of edge and stem lengths, respectively. We
also quantify via cross-correlation the contact distribution function $H(r)$
in Fig. 3(c), defined as the probability that the minimum distance from a
random position in the Al matrix to the Si tree structure is less than $r$ for
$r>0$ [26]. Superimposed are the same descriptors from Ref. [26] on the
impurity-modified eutectic. Unsurprisingly, both edge and stem lengths of the
impurity-modified structure are on the order of micrometers, while the
quenched structure shows finer features down to the nanometer length-scale. In
fact, we find an average fiber radius of 22.5 nm and an average fiber spacing
$\lambda$ of 91.0 nm for the quenched structure using the auto-correlation
function (see Fig. S4). This is near the finest possible spacing of 39.4 nm
predicted by Ref. [42] as the limit of steady-state eutectic growth, see Table
2. Finally, we report a surface density of 58.0307 $\mu$m-1, over 100 times
larger than that of the impurity-modified sample. Table 1 summarizes the data
from both eutectic structures.

In general, the nanoscale features of Si fibers in the LSR meltpool can be
attributed to the deep undercooling and high growth velocity. Simulations [43]
and experiments [44, 45, 46] have shown that a larger undercooling results in
“facet breaking” which, for eutectics, should coincide with a transition from
flake-like to fibrous Si. Adopting the undercooling-spacing-velocity
relationships modified for rapid solidification [14], we estimate a growth
velocity of 30.2 mm s-1 (which, fortuitously, corresponds to the laser
scanning speed) and an undercooling of 72.7 K. In comparison, Hosch and
coworkers [19, 47] reported that the flake-to-fiber transition of Si occurs in
two stages with the first being the in-plane break-down of Si plates and the
second being the out-of-plane Si branch growth. This conclusion was drawn from
deep etching experiments; as such, it is difficult to ascertain whether fiber
growth above the second transition is truly 3D in nature or is confined within
the Si plates, as the authors themselves acknowledge [19]. Nevertheless, our
calculated velocity is far above that of the second transition (see Table 2),
which may explain the fibrous structure in Fig. 2c-d.

| $f_{\mathrm{Si}}$ | $\mathrm{wt}\%\mathrm{Si}$ | $\lambda(\mu\mathrm{m})$ | $s_{S}\left(\mu m^{-1}\right)$ | $r_{eq}(\mu\mathrm{m})$ | Ref.
---|---|---|---|---|---|---
Quench-modified | $0.1805$ | ${\color[rgb]{0,0,0}20}$ | $0.091$ | $58.0307$ | $0.0225$ | This work
Impurity-modified | $0.14$ | ${\color[rgb]{0,0,0}7}$ |  $\sim 2$ * | $0.527$ | $0.55$ | [26]
DLA |  $0.3512$ ** | ${\color[rgb]{0,0,0}36}$ | $\mathrm{n}/\mathrm{a}$ | $\mathrm{n}/\mathrm{a}$ | $\mathrm{n}/\mathrm{a}$ | This work
Eutectic point (nominal) | $0.143$ | ${\color[rgb]{0,0,0}12.6}$ | $\mathrm{n}/\mathrm{a}$ | $\mathrm{n}/\mathrm{a}$ | $\mathrm{n}/\mathrm{a}$ | [48]
Table 1: Basic descriptors of eutectic microstructure produced by different
routes: $f_{Si}$ is the volume fraction of Si phase (directly measured from
the segmented volume, in our work), wt% is weight percent, $\lambda$ is the
fiber spacing (defined as the distance between neighboring stems of Si),
$s_{S}$ is the specific surface density of Si phase (the surface area of Si
phase divided by its volume), and $r_{eq}$ is the radius of Si fibers. Based
on past reports [49], $\lambda$ does not vary with composition, holding all
else constant.
*: Value is calculated from the contact distribution function $H(r)$ provided in Fig. 12 of Ref. [26].
**: The relatively large Si phase fraction obtained from DLA simulation is caused by dilating the tree structure to match the 22.5 nm fiber radius from the quenched sample. |  $\lambda$ $(\mathrm{nm})$ | V $(\mathrm{mm}/\mathrm{s})$ |  $\Delta$T (K) | Ref
---|---|---|---|---
Rapid quenching | $91.0$ | 30.2 | $72.7$ | This work
Coupled zone boundary, for 16 wt%Si | $65.3$ | $46.2$ | $120.6$ | [14]
Limit of fiber spacing | $39.4$ | 133 | $240.2$ | [42]
Upper limit of flake-to-fiber transition |  $11-15\times 10^{3}$ *** | $0.5-0.95$ | $\sim 10^{*}$ | [19]
Table 2: Solidification parameters for this study (top row) compared with
other critical values from the literature (below). ***: Value is estimated
from the critical velocity obtained in Ref. [19].

Aside from the obvious differences in length-scale, it remains to be
determined whether the two processing routes lead to the same morphology of
the Si phase. To answer this question, we plot the three histograms again,
this time normalizing the length-scales to a range between zero and one (see
Fig. 3(d-f)). The difference in the two distributions remains even after
controlling for length-scale, indicating two fundamentally different
morphologies with different origins. We conduct a two-sample
Kolmogorov–Smirnov (K-S) test to quantify the statistical distance between the
two distributions [50, 51, 52] and ultimately reject the null hypothesis that
the two curves are drawn from the same distribution, at a 5% significance
level. It follows that the two microstructures are not self-similar.

Figure 3: Quantitative comparison of the tree-like structures according to
three metrics: (a,d) edge lengths, (b,e) stem lengths, and (c,f) contact
distributions. Data presented on histograms come from quenched- (red) and
impurity-modified (blue) structures. (a-b) are plotted on a log-linear scale
due to the significant difference in length scale between the two. Top row
shows length-scales in physical coordinates while bottom row is normalized to
control for differences in length-scale. On the bottom plots we give also
distributions obtained from DLA simulation (yellow) using a sticking
coefficient of 0.4, see text for details.

Instead, we posit that rapid cooling in LSR not only produces a much finer
eutectic microstructure, but also a highly interconnected and isotropic
branching behavior. To test this idea more rigorously, we compared
stereographic projections of the Si fiber edge directions in the quenched
structure against a theoretical prediction from multiple twinning, see Figs.
4. The latter is determined by calculating the total number of unique crystal
orientations $N(n)$ that are possible after $n$th-order twinning on {111}
planes, which follows a geometric series [1, 53]:
$N(n)=4\sum_{t=1}^{n}3^{t-1}+1$. For example, after $n=8$ repeated twinning
events, there are a total of $N(8)=13121$ unique crystal orientations (and
hence growth directions) that cover the $\langle 100\rangle$ stereographic
projection, although not uniformly. That is, Fig. 4(b) shows a preferred four-
fold symmetry, which reflects the cubic symmetry of the Si crystal. In
contrast, the fiber growth directions are non-directional or isotropic in Fig.
4(a).

We rationalize the above observations by noting that more growth sites are
operative at high undercooling that are otherwise inaccessible at lower
undercooling [54]. As more and more sites become active, the kinetic factor of
Si in the revised Wilson-Frenkel law [55] will approach that of typical
metals. That is, provided that the undercooling is above the threshold for the
flake-to-fiber transition (as it is in our case), growth on multiple sites
will naturally lend itself to a more isotropic morphology (namely, branching
in any and all directions irrespective of the underlying crystallography of
Si). An alternative explanation is that the kinetic undercooling of the Si
phase increases with growth rate, such that the lead distance of the Si phase
at the growth front would be gradually reduced. Eventually the Al and Si
phases would be fully coupled at the growth front, such that the shape of the
Si phase would no longer be governed by faceted growth [56, 57].

In contrast, theories for impurity-induced modification argue that the
impurity atoms (such as Na or Sr) are rejected by the solidification front;
accumulate at the eutectic-liquid interface; interfere with the layer growth
of Si (once the eutectic Si has grown far enough to collect sufficient
impurity atoms); and increase the number of twin defects in Si, thus enabling
more frequent branching [58, 56, 7]. This cycle of crystallographic branching
occurs with a characteristic frequency that is given by the edge length in
Fig. 3(a,d). That said, branching is still constrained by the favored growth
directions of Si and the locations of the twin plane grooves (provided the
alloy is solidified under low growth rates) [59, 60, 20].

Figure 4: Distributions of Si fiber growth directions: comparison between (a)
experimental result on ultra-fast quenching and (b) theoretical prediction of
the branched morphology caused by multiple twinning on {111} planes. Zone axis
(black dot) in stereographic projections is specimen $z$ (cf. Fig. 1) and
crystallographic $\langle 100\rangle$ direction, respectively. Cross-hatched
regions include an absence of data.

We put the above result in a broader context by comparing, for the first time,
the morphology of Si to another, prototypical branched structure produced by
diffusional aggregation: a random fractal [61, 62]. We follow standard
algorithms for DLA [63, 61] to simulate the latter and test for its similarity
to Si morphology under quenching. We restrict the movement of random walkers
along the specimen $z$ direction to match the growth direction of the eutectic
in experiment. We use the K-S test again to determine the sticking coefficient
(the probability that a walker will stick to the aggregate) that resembles our
quenched tree structure most closely, see Fig. S5. Then, as before, we obtain
statistics on edge length, stem length and contact distribution (yellow curves
in Fig. 3(d-f)).

According to the K-S test, the edge lengths of the random fractal and the
quench-modified structure come from the same distribution at a 5% significance
level. We reject the null hypothesis on stem length, however, although we
suggest that this discrepancy is because the stem length is sensitive to the
volume of the colonies from which the distributions are obtained. Likewise,
the contact distributions of the two structures show a clear difference. This
is due to the physics of directional eutectic solidification.555See Ref. [64]
for the multi-particle extension of DLA for the simulation of solidification
structures. That is, with the high thermal gradient of LSR, the Si fibers are
aligned with the macroscopic growth direction (6.4% of edges are within 15∘ of
$z$ vs. 3.4% for a random distribution, see also Fig. 4(a)). This would imply
that the contact distribution $H(r)$ increases linearly with normalized
distance $r$. Taken altogether, our analysis of the descriptors indicates that
the ultrafine Si in LSR is in its own morphological class, distinct from
impurity-modified and random fractal structures.666Pursuing also a comparison
of the three histograms for the impurity-modified and random fractal
structures, we find a significant difference at the 5% confidence level
according to the K-S statistic.

In this paper, we characterize in 3D the morphology of an ultra-fine Al-Si
eutectic processed via laser rapid solidification. We map the fibrous Si
microstructure to a network of nodes and links to quantify its branching
behavior. This representation brings to light that the quench-modified
eutectic not only has a finer spacing (on the nanoscale) compared to its
impurity-modified counterpart, but perhaps more surprisingly, that the
morphologies of the two eutectics are not necessarily the same. For example,
the branching behavior of the quench-modified eutectic is remarkably isotropic
(somewhat akin to a random fractal produced via DLA). This would indicate the
absence of Si facets exposed to the liquid, the result of either kinetic
roughening or coupled growth at deep undercooling. Further insight may be
gained from examining the Si morphology of impurity-doped samples under LSR,
for which experiments are well underway. Broadly, our quantitative analysis
can be used to generate synthetic eutectic microstructures that reflect the
inherent complexity of real ones.

## Acknowledgements

We gratefully acknowledge financial support from the National Science
Foundation (NSF) CAREER program under award number 1847855. Amit Misra and
Huai-Hsun Lien thank the support from DOE, Office of Science, Office of Basic
Energy Sciences, grant number DE-SC0016808 for the laser processing work. We
acknowledge assistance from Metin Kayitmazbatir and late Prof. Jyoti Mazumder
in laser scanning experiments. We thank the University of Michigan College of
Engineering for financial support and the Michigan Center for Materials
Characterization for use of the instruments and staff assistance.

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## Supplementary material

Video V1:  Visualization of the 3D fibrous nano-scale eutectic. The first-half
of the video animates the growth process from different perspectives. The
second-half shows a single tree-like structure of Si, and its associated nodes
and links, used to quantify the branching behavior (shown also in Fig. 2c-d).
Figure S1: Side-by-side visual comparison of the impurity modified [20] and
quench modified fibrous structure of Si phase [19]. The Al phase has been
chemically removed in both images. Adapted with permission.

### Procedure for tracing colony boundaries

We decompose the microstructure into colonies by identifying the Si fibers
that are located along the colony boundaries. According to the arguments given
in Ref. [35], fibers along the boundaries tend to be coarser (and hence have a
larger eutectic spacing). With this in mind, we selected eight descriptors to
analyze the ensemble of fibers in the microstructure: ($i$) volume, ($ii$)
surface area, ($iii$) ratio of the largest and smallest principal axis length
in 3D, ($iv$) solidity, as well as ($v-viii$) the maximum, minimum, total, and
mean distances between fibers and all of their nearest neighbors. The nearest
neighbours are defined as those fibers closest to the target fiber in all
directions. To identify the descriptors that help to distinguish between
fibers at the boundary vs. those at the interior, we perform a principal
component analysis. The first principal component (PC1) is a combination of
volume $V$, surface area $S$, and aspect ratio $\phi$; it contributes more
than 90% of the variance. Accordingly, we define a cost function to be a
linear function of these three parameters, where the coefficients are taken as
the weights $w$ in PC1, i.e., $\mbox{cost}=w_{V}V+w_{S}S+w_{\phi}\phi$. A
higher cost value indicates that the Si fiber is located along the periphery
of the colony.

We can color each Si fiber according to its cost in the cross-sectional images
($x$-$y$ plane). The outline of the colonies can then be reasonably identified
from a projection along the image depth ($z$) for “chunks” of 30 cross-
sectional images, see one such projection in Fig. S2.

Figure S2: 2D slice from the 3D dataset, wherein the Si fibers are colored
according to value of cost function.

### Procedure for identifying tree-like structures

To subdivide the colonies into “tree”-like structures of connected fibers, we
developed two empirical rules that are consistent with the physics of
directional eutectic solidification. These rules enabled us to break links
that connected two trees (due to merging of branched fibers in solidification
[65]) and that ultimately rendered a single connected component (instead of
two separate ones). Fig S3 provides a schematic illustration of both rules:

  1. 1.

Rule #1: No backwards growth. Here, we break links that point backwards (along
$-z$), i.e., away from the macroscopic growth direction ($+z$). Physically, it
is not possible for a fiber to grow into a region that has already solidified.

  2. 2.

Rule #2: First come, first occupy. Here, we consider nodes that are shared
between two or more trees. We assign them to the tree that arrives at that
node earlier than the others, i.e., the tree that ‘wins’ is likely the one
that has a shorter path from the starting seed to the shared node. The path
length is calculated as the total length of the previous links from the
starting seed up to the shared node. This rule follows from the fact that Si
fibers can impinge upon fibers that were solidified beforehand.

While these rules can also impact the final tree morphology of Si phase, we
try to minimize such an effect by selecting out those trees making their way
through the whole colony as our target structures when gathering the branching
statistics.

Figure S3: Schematic of the two empirical rules that were enforced in order to
separate the tree structures in the quenched sample. Here, $z_{i}$ is the $z$
coordinate of node $i$ (along the growth direction) and $l_{i}$ is the length
of the path (in black) between seed $i$ and the shared node. See text for
details.

### Basic descriptors of microstructure

We obtain the Si fiber diameter and spacing by calculating the 2D auto-
correlation of Si phase. Fig. S4(a) shows a section of one 2D slice that we
use to demonstrate our procedure, taken from the center of a colony. Fig.
S4(b) shows the auto-correlation of Fig. S4(a). The size of the central peak
(in yellow, a.u.) indicates the average size of the Si fibers. The distance
between the central peak and the first-order ring indicates the spacing
between the Si fibers. Note that it is invariant with direction, indicating
that the eutectic microstructure is spatially isotropic. Fig. S4(c) shows the
radial distribution function, computed from Fig. S4(b) with respect to the
distance from the center of a Si fiber. The width of the primary peak gives
the fiber diameter as 22.5 nm, and the distance between primary and secondary
peaks gives the average spacing as 91.0 nm.

Figure S4: Determination of basic statistics by auto-correlation function. (a)
Section of one representative 2D slice from the 3D dataset. (b) Density plot
of the auto-correlation (a.u.). (c) Radial distribution function, where the
origin corresponds to the center of a Si fiber. Accordingly, the average fiber
radius and average spacing in this example is 22.5 nm and 91.0 nm,
respectively.

### DLA simulation

When conducting DLA simulation, the sticking coefficient is one input
parameter that influences strongly the structure of the tree clusters formed
by random walkers [61, 62]. A sticking coefficient of 1 implies that all
walkers will stick to the cluster, and 0 the converse. To determine which
coefficient to use, we vary the coefficient from 0.1 to 1 in increments of 0.1
and generate 20 sets of random fractal clusters for each case. We then obtain
branching statistics of all DLA tree structures following the same procedure
(skeletonization) that we followed for the experimental data. We again use the
K-S statistic to codify the similarity of distributions of edge and stem
length between DLA and experimental data on the quench-modified structure. The
result is shown in Fig. S5 where a lower K-S statistic value indicates a
closer agreement in branching statistics. With this data in hand, we select a
sticking coefficient of 0.4 to match the experimental data as best as
possible.

Figure S5: Similarity measures between the quenched microstructure vs. DLA
structures that were simulated with varying stick coefficients. The
coefficient of 0.4 corresponds most closely to the quenched structure visually
and quantitatively by the K-S test, and is the one that we use in our analysis
in Fig. 3(d-f). Insets show examples of tree-like structures generated via DLA
using different sticking coefficients.


Applications of a class of transformations of complex sequences

John M. Campbell

###### Abstract

Through an application of a remarkable result due to Mishev in 2018 concerning
the inverses for a class of transformations of sequences of complex numbers,
we obtain a very simple proof for a famous series for $\frac{1}{\pi}$ due to
Ramanujan. We then apply Mishev’s transform to provide proofs for a number of
related hypergeometric identities, including a new and simplified proof for a
family of series for $\frac{1}{\pi}$ previously obtained by Levrie via
Fourier–Legendre theory. We generalize this result using Mishev’s transform,
so as to extend a result due to Guillera on a Ramanujan-like series involving
cubed binomial coefficients and harmonic numbers.

MSC Classification: 33C20, 11Y60

Key words: Ramanujan series, closed form, Ramanujan-type series,
hypergeometric series

##  1 Introduction

Included in Ramanujan’s first letter to Hardy was the now-famous formula

$\frac{2}{\pi}=\sum_{n=0}^{\infty}\left(-\frac{1}{64}\right)^{n}\binom{2n}{n}^{3}(4n+1)$
(1)

that had been previously proved by Bauer in 1859 [6] via a Fourier–Legendre
expansion; see also [8, 12, 22], along with [5] and the references pertaining
to (1) therein. As suggested in [19], Ramanujan’s series evaluations as in (1)
are of interest in part because of the close association of such formulas with
advanced subjects in number theory and complex analysis. In 1994, Zeilberger
[13] used the _Wilf–Zeilberger_ (WZ) method [26] to produce a very simple and
elegant computer proof of (1). In this article, we present a new, one-page
proof of (1) that does not involve the WZ method. This new proof of ours
relies on a remarkable transformation identity for complex sequences
introduced in 2018 by Mishev in what we consider to be an overlooked and
underrated research article [24] that does not seem to have been yet cited.
Furthermore, we apply Mishev’s transform to construct a simplified proof for
and a generalization of an identity due to Levrie [22] on an infinite family
of series for $\frac{1}{\pi}$ generalizing (1), and we apply our
generalization of Levrie’s identity to extend a remarkable result due to
Guillera [15] on a Ramanujan-like series involving harmonic numbers and cubed
central binomial coefficients. Letting
$H_{m}=1+\frac{1}{2}+\cdots+\frac{1}{m}$ denote the $m^{\text{th}}$ harmonic
number, one of the main results in this article is our Ramanujan-like series
formula

$-\frac{2\ln
2}{\pi}=\sum_{n=0}^{\infty}\left(-\frac{1}{64}\right)^{n}\binom{2n}{n}^{3}(4n+1)H_{2n},$
(2)

noting the resemblance to Ramanujan’s formula in (1).

###  1.1 Mishev’s transform

We will use the Pochhammer symbol
$(\alpha)_{n}=\alpha(\alpha+1)\cdots(\alpha+n-1)$ for $n\in\mathbb{N}_{0}$ and
the usual notation for generalized hypergeometric series:

${}_{p}F_{q}\left[\begin{matrix}a_{1},a_{2},\ldots,a_{p}\\\
b_{1},b_{2},\ldots,b_{q}\end{matrix}\ ;\
x\right]=\sum_{n=0}^{\infty}\frac{\left(a_{1}\right)_{n}\left(a_{2}\right)_{n}\cdots\left(a_{p}\right)_{n}}{\left(b_{1}\right)_{n}\left(b_{2}\right)_{n}\cdots\left(b_{q}\right)_{n}}\frac{x^{n}}{n!}.$

We also write

$\left[\begin{matrix}\alpha_{1},\alpha_{2},\ldots,\alpha_{\ell_{1}}\vspace{1mm}\\\
\beta_{1},\beta_{2},\ldots,\beta_{\ell_{2}}\end{matrix}\right]_{n}=\frac{\left(\alpha_{1}\right)_{n}\left(\alpha_{2}\right)_{n}\cdots\left(\alpha_{\ell_{1}}\right)_{n}}{\left(\beta_{1}\right)_{n}\left(\beta_{2}\right)_{n}\cdots\left(\beta_{\ell_{2}}\right)_{n}}.$

The main transformation identity in [24] may be formulated in an equivalent
way as below.

######  Theorem 1.

(Mishev, 2018) For a sequence $(x_{n}:n\in\mathbb{N}_{0})$ of complex numbers,
and for a parameter $a$ in $\mathbb{C}\setminus\\{0,-1,-2,\ldots\\}$, the
equality

$x_{n}=\frac{1}{n!(a+1)_{n}}\sum_{k=0}^{n}\left[\begin{matrix}a,1+\frac{a}{2},-n\vspace{1mm}\\\
1,\frac{a}{2},1+a+n\end{matrix}\right]_{k}\sum_{\ell=0}^{k}(-k)_{\ell}(k+a)_{\ell}x_{\ell}$
(3)

holds true (Theorem 3.1 in [24]).

This is proved in [24] by defining a transform
$L_{a}(x)_{n}=\sum_{k=0}^{n}(-n)_{k}(n+a)_{k}x_{k}$ for a complex-valued
sequence $(x_{n})_{n\in\mathbb{N}_{0}}$, and by determining the inverse of
this transform using Dixon’s classical summation theorem.

##  2 A new and simple proof of Ramanujan’s formula

Zeilberger’s proof of (1) relied on a finite hypergeometric sum identity that
was established via a computer-generated WZ proof certificate, and Carlson’s
theorem [2, p. 39] [30] was used to establish that the corresponding identity
obtained by replacing the upper limit of the aforementioned finite sum with
$\infty$ holds for suitably bounded real $n$, with (1) as a special case of
this resultant infinite sum identity. It is unclear as to how the WZ proof
from [13] may be altered so as to obtain our new results given in Section 3.

It is well-known [4, 9, 10, 12, 21] that Ramanujan’s formula, as in (1), may
be proved via the limiting behaviour of a well-poised ${}_{5}F_{4}(1)$-series
given by Dougall. In contrast, our new proof of (1) relies on a
${}_{4}F_{3}(-1)$-identity that we obtain using Mishev’s work [24].

Another way of proving (1) would be given by expressing the generating
function for the sequence of cubed central binomial coefficients via Clausen’s
hypergeometric product (cf. [3]), so as to obtain an expression involving the
square of the complete elliptic integral of the first kind, and by then using
an elliptic singular value, and some classical identities for the complete
elliptic integrals. This sophisticated approach is in contrast to our brief
proof of Theorem 2 below. Yet another proof of (1) was given in [23] using
recurrences and reindexing arguments for a family of ${}_{3}F_{2}(1)$-series,
in contrast to our derivation of the ${}_{4}F_{3}(-1)$-identity in (5).

Yet another proof of (1) is given in [25] using a series acceleration
technique due to Gosper applied to a classical infinite product identity from
Euler. Our proof of (1) below is dramatically different and considerably
simpler.

######  Theorem 2.

Ramanujan’s formula for $\frac{1}{\pi}$ shown in (1) holds true.

###### Proof.

We set $x_{\ell}=\frac{1}{(\ell!)^{2}}$ in Mishev’s transformation identity,
as in Theorem 1. Since $k\geq 0$ is an integer, we see that we may then
rewrite the inner sum in (3) as

$\sum_{\ell=0}^{\infty}\frac{(-k)_{\ell}(k+a)_{\ell}}{\left(\ell!\right)^{2}},$
(4)

since $(-k)_{\ell}$ vanishes for $k<\ell$. According to Gauss’ famous
${}_{2}F_{1}(1)$-identity

${}_{2}F_{1}\left[\begin{matrix}a,b\\\ c\end{matrix}\ ;\
1\right]=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)},$

we see that the infinite series in (4) evaluates as
$\frac{(-1)^{k}(a)_{k}}{k!}$. So, this immediately gives us that Mishev’s
identity, as in (3), for the special case whereby
$x_{\ell}=\frac{1}{(\ell!)^{2}}$, gives us that

${}_{4}F_{3}\\!\\!\left[\begin{matrix}a,a,-n,1+\frac{a}{2}\vspace{1mm}\\\
1,\frac{a}{2},n+a+1\end{matrix}\ ;\ -1\right]=\frac{(a+1)_{n}}{n!},$ (5)

for nonnegative integers $n$ and for suitably bounded $a$ (cf. [13]). As in
[13], by Carlson’s theorem, we find that (5) also holds for suitably bounded
non-integer values for $n$. In particular, setting $a=\frac{1}{2}$ and
$n=-\frac{1}{2}$, this immediately gives us that

${}_{4}F_{3}\\!\\!\left[\begin{matrix}\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{5}{4}\vspace{1mm}\\\
\frac{1}{4},1,1\end{matrix}\ ;\ -1\right]=\frac{2}{\pi},$ (6)

and this is precisely the same as Ramanujan’s formula in (1), rewriting the
summand of the left-hand side of (6) using central binomial coefficients. ∎

The above application of Mishev’s transform does not appear in [24], and does
not seem to be present in any relevant literature. As stated above, Mishev’s
article [24] does not seem to have been cited previously, as of this note
being written. This, together with our above application of Theorem 1,
suggests that Mishev’s 2018 article in [24] is underappreciated. It also
appears that no equivalent or meaningfully similar formulation of our proof of
(1) has appeared in any in any past literature. For research relevant to
Zeilberger’s WZ proof in [13], we refer to Guillera’s work in WZ theory [14,
16, 17] and past work on supercongruences concerning Ramanujan-type sums [20,
28, 31].

##  3 Further results

The hypergeometric identity in (5) is of interest in its own right, and
motivates the use of Mishev’s transform [24] with inputted sequences given by
variants of the $x$-function involved in our proof of Theorem 2.

######  Example 1.

Setting $x_{\ell}=\frac{1}{\ell!(\ell+1)!}$ in the formulation of Mishev’s
transform given in Theorem 1, we may obtain that

${}_{4}F_{3}\\!\\!\left[\begin{matrix}a,a-1,-n,1+\frac{a}{2}\vspace{1mm}\\\
2,\frac{a}{2},n+a+1\end{matrix}\ ;\
-1\right]=\frac{\left(a+1\right)_{n}}{(n+1)!}$

for suitably bounded $a$ and $n$; we may again apply Carlson’s theorem, as
above.

######  Example 2.

Setting $x_{\ell}=\frac{1}{\ell!\left(\ell+\frac{1}{2}\right)!}$, Mishev’s
transform gives us that

${}_{4}F_{3}\\!\\!\left[\begin{matrix}a,-n,\frac{a}{2}+1,a-\frac{1}{2}\vspace{1mm}\\\
\frac{3}{2},\frac{a}{2},a+n+1\end{matrix}\ ;\
-1\right]=\frac{\left(a+1\right)_{n}\,\sqrt{\pi}}{2\left(n+\frac{1}{2}\right)!}$

for suitably bounded $a$ and $n$. For example, we may obtain the exotic
hypergeometric series evaluation by setting $a=\frac{1}{4}$ and
$n=\frac{1}{4}$:

${}_{4}F_{3}\\!\\!\left[\begin{matrix}-\frac{1}{4},-\frac{1}{4},\frac{1}{4},\frac{9}{8}\vspace{1mm}\\\
\frac{1}{8},\frac{3}{2},\frac{3}{2}\end{matrix}\ ;\
-1\right]={\frac{2\,\sqrt{2}}{3}}.$

We find it appropriate to highlight the following evaluation for a
${}_{4}F_{3}(-1)$-series with three real parameters as a theorem.

######  Theorem 3.

The evaluation

${}_{4}F_{3}\\!\\!\left[\begin{matrix}a,-n,\frac{a}{2}+1,a-b+1\vspace{1mm}\\\
b,\frac{a}{2},1+a+n\end{matrix}\ ;\
-1\right]=\left[\begin{matrix}a+1\vspace{1mm}\\\ b\end{matrix}\right]_{n}$

holds true for suitably bounded real $a$, $b$, and $n$.

###### Proof.

This follows in a direct way from Mishev’s transform, as given in Theorem 1,
by setting $x_{\ell}=\frac{1}{\left(1\right)_{\ell}\left(b\right)_{\ell}}$ for
a real parameter $b$, and by then applying Carlson’s theorem. ∎

By again setting $a=\frac{1}{2}$ and $n=-\frac{1}{2}$, recalling our proof of
Ramanujan’s formula in (1), Theorem 3 then gives us an infinite family of
generalizations of this famous formula.

###  3.1 An identity due to Levrie

We offer a simplified proof and a generalization of the following identity
that had been introduced by Levrie and proved in [22] via Fourier–Legendre
theory. The below proof is remarkably simple and different compared to
Levrie’s proof, noting that we do not require orthogonal polynomials,
integration identities for Legendre polynomials, etc.

######  Theorem 4.

(Levrie, 2010) The identity

$\displaystyle\sum_{i=0}^{\infty}(-1)^{i}\frac{\left(\frac{1}{2}\right)_{i}^{3}}{(i!)^{3}}\frac{4i+1}{(2i-(2k-1))\cdots(2i-1)(i+1)\cdots(i+k)}$
$\displaystyle=(-1)^{k}\frac{1}{2^{k-1}\left(\frac{1}{2}\right)_{k}^{2}}\frac{1}{\pi}$

holds for all natural numbers $k$ [22].

###### Proof.

Again, we set $a=\frac{1}{2}$ and $n=-\frac{1}{2}$ in Theorem 3, giving us
that

${}_{4}F_{3}\left[\begin{matrix}\frac{1}{2},\frac{1}{2},\frac{5}{4},\frac{3}{2}-b\vspace{1mm}\\\
\frac{1}{4},1,b\end{matrix}\ ;\
-1\right]=\frac{2\,\left(b-\frac{1}{2}\right)_{\frac{1}{2}}}{\sqrt{\pi}}$ (7)

for suitably bounded real $b$. For the special case whereby $b$ is a natural
number, and letting $i\in\mathbb{N}_{0}$, we find that

$\frac{\left(\frac{3}{2}-b\right)_{i}}{\left(b\right)_{i}}$ (8)

may be written as

$\displaystyle(-1)^{b+1}\frac{\left(\frac{(2(b-1))!}{2^{b-1}}\right)}{(2i-1)\cdots(2i-2(b-1)+1)(i+1)\cdots(i+b-1)}\frac{\binom{2i}{i}}{4^{i}}.$

Rewriting the summand on the left-hand side of (7) accordingly, we obtain an
equivalent form of the desired identity due to Levrie. ∎

We may regard (7) as generalizing Levrie’s identity, as given in Theorem 4,
for real values $b$.

###  3.2 Extensions of a series evaluation due to Guillera

A remarkable Ramanujan-like series due to Guillera [15] that involves harmonic
numbers is reproduced in an equivalent form as below, noting the close
resemblance to Ramanujan’s series (1):

$\frac{1}{6\sqrt{2}\pi}\left(\frac{\Gamma\left(\frac{1}{8}\right)\Gamma\left(\frac{3}{8}\right)}{\Gamma\left(\frac{1}{4}\right)\Gamma\left(\frac{3}{4}\right)}\right)^{2}-\frac{4\ln(2)}{\pi}=\sum_{n=0}^{\infty}\left(-\frac{1}{64}\right)^{n}\binom{2n}{n}^{3}(4n+1)H_{n}.$
(9)

This is proved in [15] using the identity

$\frac{1}{2}\sum_{n=0}^{\infty}(-1)^{n}\frac{\left(\frac{1}{2}+x\right)^{3}_{n}}{\left(1\right)_{n}^{3}}(1+2x+4n)=\frac{\cos\pi
x}{\pi}$

obtained in [10] together with an expansion for

$\frac{1}{2}\sum_{n=0}^{\infty}(-1)^{n}\frac{\left(\frac{1}{2}+x\right)^{3}_{n}}{\left(1+x\right)_{n}^{3}}\left(1+4x+4n\right)$

offered in [18]. It is unclear as to how this kind of approach may be
generalized so as to evaluate series involving

$\left(\frac{1}{64}\right)^{n}\binom{2n}{n}^{3}\Delta(n)$ (10)

more generally, letting $(\Delta(n):n\in\mathbb{N}_{0})$ denote a sequence of
harmonic-type numbers. This has motived our recent work in [8] on the use of
the WZ method in the evaluation of a Ramanujan-inspired series involving (10).
However, it is unclear as to how the methodologies from [8, 15] may be applied
to evaluate the series obtained by replacing the harmonic factor $H_{n}$ in
(9) with, say, $H_{2n}$ or $O_{n}=1+\frac{1}{3}+\cdots+\frac{1}{2n-1}$. We
have successfully solved this problem, through an application of our
generalization, as in (7), of Levrie’s identity. It appears that known
integration and coefficient-extraction techniques from References as in [7,
11, 29] for evaluating series involving

$\left(\frac{1}{16}\right)^{n}\binom{2n}{n}^{2}\Delta(n)$

cannot be applied to the much more difficult problem of evaluating harmonic
sums involving _cubed_ , as opposed to squared, central binomial coefficients.
This emphasizes the remarkable nature about our results as in Theorem 5 below.

Note that the digamma function is such that the following holds (cf. [27],
§9):

$\psi(z)=\frac{d}{dz}\ln\Gamma(z)=\frac{\Gamma^{\prime}(z)}{\Gamma(z)}=-\gamma+\sum_{n=0}^{\infty}\frac{z-1}{(n+1)(n+z)},$
(11)

with $\gamma$ the Euler–Mascheroni constant.

######  Theorem 5.

The Ramanujan-like series evaluation

$-\frac{2\ln
2}{\pi}=\sum_{i=0}^{\infty}\left(-\frac{1}{64}\right)^{i}\binom{2i}{i}^{3}(4i+1)H_{2i}$

holds true.

###### Proof.

Applying $\frac{d}{db}$ to the summand factor highlighted in (8), we obtain:

$\displaystyle-{\frac{1}{\left(b\right)_{i}}\left({\frac{3}{2}}-b\right)_{i}\left(\psi\left(i+{\frac{3}{2}}-b\right)-\psi\left({\frac{3}{2}}-b\right)\right)}-$
$\displaystyle{\frac{\psi\left(i+b\right)-\psi\left(b\right)}{\left(b\right)_{i}}\left({\frac{3}{2}}-b\right)_{i}}.$

So, applying term-by-term differentiation to both sides of our generalization
of Levrie’s identity, with respect to the free variable in (7), and then
setting $b=1$, this gives us that the sum of all expressions of the form

${\frac{\left(4\,i+1\right)\left(-1\right)^{i+1}}{{\pi}^{{\frac{3}{2}}}\left(i!\right)^{3}}\left(\psi\left(i+{\frac{1}{2}}\right)+2\,\gamma+2\,\ln\left(2\right)+\psi\left(i+1\right)\right)\left(\left(i-{\frac{1}{2}}\right)!\right)^{3}}$

for $i\in\mathbb{N}_{0}$ admits the following closed-form evaluation:
$4\,{\frac{\ln\left(2\right)}{\pi}}$. We may find that
$\psi\left(i+\frac{1}{2}\right)=2O_{i}-\gamma-2\ln 2$, in view of (11). This,
together with the equality $\psi(i+1)=H_{i}-\gamma$, give us that

$\sum_{i=0}^{\infty}(-1)^{i+1}(2O_{i}+H_{i})(4i+1)\frac{\left(\left(i-\frac{1}{2}\right)!\right)^{3}}{\pi^{3/2}(i!)^{3}}=4\,{\frac{\ln\left(2\right)}{\pi}}.$

Making use of the Legendre duplication formula [1, p. 256], this gives us that

$\sum_{i=0}^{\infty}\left(-\frac{1}{64}\right)^{i}\binom{2i}{i}^{3}(4i+1)(2O_{i}+H_{i})=-4\,{\frac{\ln\left(2\right)}{\pi}}.$

In view of the relation $O_{i}=H_{2i}-\frac{1}{2}H_{i}$, we obtain the desired
result. ∎

We see that the above Theorem, in conjunction with Guillera’s formula in (9),
also allows us to evaluate both

$\sum_{i=0}^{\infty}\left(-\frac{1}{64}\right)^{i}\binom{2i}{i}^{3}(4i+1)H_{2i}^{\prime}$

and

$\sum_{i=0}^{\infty}\left(-\frac{1}{64}\right)^{i}\binom{2i}{i}^{3}(4i+1)O_{i},$

letting $H_{n}^{\prime}=1-\frac{1}{2}+\cdots+\frac{(-1)^{n+1}}{n}$ denote the
$n^{\text{th}}$ alternating harmonic number. It appears that Abel-type
summation lemmas, following the techniques from [8], cannot be used to
evaluate the above summations.

We are actively involved in exploring the application of Mishev’s transform in
conjunction with WZ theory, and through the use of the modified Abel lemma on
summation by parts. Also, by setting
$x_{\ell}=\frac{y^{\ell}}{(1)_{\ell}^{2}}$ in Theorem 1 for a variable $y$,
this gives us that the inner sum in (3) may be written as a Jacobi polynomial;
this motivates the application of Mishev’s transform using the theory of
orthogonal polynomials.

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John M. Campbell

York University

Toronto, Ontario, Canada

<EMAIL_ADDRESS>