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Sensors and Materials, Vol. 35, No. 10 (2023) 4617–4632
MYU Tokyo
S & M 3418

Analysis of Chemical Composition Formula of Sealing Alloys
Using Nearest-neighbor Two-shell Model
Hai-Lian Hong, Chi-Hsin Yang,* Hui-Xian Yan,
Lian-Hui Zheng, and Kun-Chieh Wang
School of Mechanical and Electric Engineering, Sanming University, Sanming 365004, Fujian Province, China(Received July 14, 2023; accepted October 5, 2023)

Keywords: nearest-neighbor two-shell structure, Fe–Ni alloys, sealing alloys, structural model,
composition analysis, Friedel spherical periodic resonanceThe composition design of sealing alloys still inevitably relies on the trial-and-error method,
leading to the unnecessary waste of time and materials. The main goal of this study is tointroduce an electronic structure model, named the nearest-neighbor two-shell (NNTS) model,
to guide the composition design of the face-centered cubic (fcc) solid solution alloy. The chemical
structure formula of the model is [X–Y12]Z2, where X, Y, and Z are the composited atomics in thealloy. The atomic structure of the model is developed by comparing with the Friedel sphere
periodic electronic resonance structure. We applied the proposed model to analyze various iron–nickel (Fe–Ni) binary and ternary alloys, and the obtained results were consistent with those
calculated using the Cowley parameters. The discovery of the model is conducive to gaining a
deeper understanding of the composition distribution of alloys and also provides a new way todesign the alloy composition.

1.

Introduction

Recently, various vacuum devices have been widely used in various fields, such as microelectro-mechanical systems (MEMS),(1) solar vacuum tubes,(2) and vacuum glass.(3) To improve
the strength, antipollution performance, and service life of the vacuum devices, it is necessary toapply some sealing techniques to construct a vacuum chamber. The technology consisting of
sealing materials and sealing techniques is the core in the preparation of vacuum devices. For
example, nanomaterial graphene with a single-atom thickness is used to achieve high sealing
performance in MEMS,(4) glass is sealed with organic compounds at low sealing temperatures,(3)and various types of sealing glass with good chemical compatibility are sealed with tempered
glass.(5) Because the metal sealing material has a low melting point and excellent corrosion andoxidation resistances, it is the preferred material for sealing glass. Indium alloy is used to seal
glass edges at a low temperature of 220 ℃,(6) silver paste is used to seal tempered glass at 450 ℃
for 10 min,(7) and aluminum and glass are reliably sealed at 400 ℃ using anodic bonding
technology.(8)*Corresponding author: e-mail: 20190207@fjsmu.edu.cn

https://doi.org/10.18494/SAM4581

ISSN 0914-4935 © MYU K.K.
https://myukk.org/

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Sensors and Materials, Vol. 35, No. 10 (2023)

Iron–nickel (Fe–Ni) alloys have high strength, high thermal conductivity, and even nomagnetism. Below the temperature of 500 ℃, the most outstanding advantage is that Fe–Ni
alloys have a stable thermal-expansion coefficient that is very close to that of glass. Therefore,
Fe–Ni alloys are the suitable sealing materials to be used with glass. However, the composition
range of Fe–Ni sealing alloys is 42 to 54 wt.% nickel and is far from the 36 wt.% of the alloy withthe lowest thermal-expansion coefficient. The selection rules for the nickel content in Fe–Ni
alloys are not yet clearly indicated in the literature.
Excellent properties of alloys are governed by stable structures, which originate from shortrange ordered structures.(9–13) Mechanical properties such as strength,(14) fracture toughness,(15)and the functional properties of expansion(16) and corrosion resistance(17) are all related to theshort-range ordered structure of an alloy. Therefore, there are many studies on short-rangeordered structures, especially the nearest-neighbor two-shell (NNTS) model, which will be
proposed herein. For example, the average concentration,(18) spin wave,(19) and orderingtransition temperature(20) are all evaluated by applying the NNTS model. The computed results
are more consistent with the experimental values than those obtained by only applying thenearest-neighbor first-shell atomic model.
Motivated by the correlation between the NNTS structure and the properties of alloys, there
is an urgent need to establish the NNTS model to analyze the composition of the alloys and thento achieve an effective method for designing the alloys. The novelty and contribution of thiswork are the development of the NNTS electronic structure model and the assignment of thechemical structural formula (CSF), so as to avoid unnecessary material waste and shorten thetime duration for alloy design in the traditional alloy development.
The rest of this paper is organized as follows. In Sect. 2, the distribution of the first andsecond nearest-neighbor atoms is obtained by analyzing the three ordered phases of Fe–Ni
alloys. Then, the probability distribution of the first and second neighbor atoms of Fe–Ni alloys
is calculated by using the short-range ordered Cowley parameters. The composition of Fe–Ni
alloys is analyzed using the ideal cluster or bi-cluster formula. In Sect. 3, the most stable shortrange ordered structure model, the 2–2 model, is found by comparing the calculated models
from the Friedel spherical periodic resonance structure. In Sect. 4, the compositions ofcommonly used Fe–Ni binary and ternary sealing alloys are analyzed using the 2–2 model.
Finally, some concluding remarks are provided.

2.

Determination of Nearest Two-shell Atomic Compositions

2.1

Nearest two-shell atomic distributions

Referring to the literature,(21) the phase diagram of Fe–Ni alloys is shown in Table 1. We havefound that there are three ordered phases with highly stable structures, Fe3Ni, FeNi, and FeNi3,
easily formed at low temperatures.(21) The two nearest-neighbor cluster formulas for the orderphases of Fe–Ni alloys are analyzed below.
(1) In the ordered phase of Fe3Ni(L12), the first nearest neighbor centered on an Fe atom iscomposed of 12 nickel atoms and the second nearest neighbor is composed of six iron atoms.

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Table 1
(Color online) Two nearest-neighbor cluster formulas for the ordered phase of Fe–Ni alloys.
Component
Ordered phase Centered around Ni atom (white sphere) Centered around Fe atom (blue sphere)
Ni at.%*

25

Fe2Ni

Ni–Ni8Fe4 –Ni6

Fe–Ni12–Fe6

50

FeNi

Ni–Fe8Ni4 –Ni6

Fe–Ni8Fe4 –Fe6

75

FeNi3

Ni–Fe12–Ni6

Fe–Fe8Ni4 –Fe6

In the former case, the first neighbor is Ni8Fe4 and the second neighbor is composed of six Ni
atoms.
(2) In the ordered phase of FeNi(L10), the first neighbor centered on a Ni atom is Fe8Ni4, and thesecond neighbor is composed of six Ni atoms. Otherwise, the first neighbor centered on an Fe
atom is Ni8Fe4 and the second neighbor is composed of six Fe atoms.
(3) In the ordered phase of FeNi3(L12), the positions of the nearest shell atoms are opposite tothose of Fe and Ni atoms in the ordered phase of Fe3Ni. That is, the Fe atoms are replaced by
Ni atoms and Ni atoms are changed to Fe atoms in Fe3Ni (L12).
Remark 1
In the face-centered cubic (fcc) lattice of the L12 structure, the face center is occupied by a
type-A atom and the vertex is occupied by a type-B atom. The L10 structure comprises analternating stack of atomic surfaces consisting of only A atoms and only B atoms along the[0 0 1] direction in the fcc lattice.(22)
Remark 2
In Table 1, at.% is the abbreviation for the atomic percentage of Ni in Fe–Ni alloys. Each of
the three ordered phases of Fe–Ni alloys similarly show the property that the first shell tends tobe composed of dissimilar atoms and the second shell tends to be composed of atoms of the same
type in the central atom of the ordered phase. This means that the attraction between dissimilar
atoms is greater than that between atoms of the same type. When dissimilar atoms are arranged
as close neighbors to each other, an Fe–Ni alloy has the lowest energy and the most stable4620

Sensors and Materials, Vol. 35, No. 10 (2023)

structure owing to the strong binding force between the atoms. In Ref. 23, it is pointed out thatthe chemical short-range ordered structure is determined by the magnitude and sign of mixing
enthalpy. When the sign of mixing enthalpy is negative, heterogeneous atoms tend to be close
neighbors, and when the sign is positive, homogeneous atoms tend to be close neighbors.
As an example, in single-phase fcc Fe–Ni alloys, the mixing enthalpy of Fe–Ni alloys with a
Ni content exceeding 25 at.% is ΔHFe–Ni = 0 to 9.6 KJ/mol.(23) The dissimilar atoms, Fe and Ni
atoms, tend to be close to each other with a negative sign of mixing enthalpy. In this case, atomsof the same type, Fe and Fe or Ni and Ni atoms, appear as the second nearest atoms. Then, theattraction between dissimilar atoms in Fe–Ni alloys is greater than that between atoms of thesame type. This is further proven later.
2.2

Ideal chemical composition formula (CCF) defined by Cowley parameters

The short-range ordered structure of solid solution alloys is characterized by Cowleyparameters.(24) Assuming that two types of atom, type-U and type-V, are found in the solution ofan alloy, the Cowley parameter is defined by α = 1 − PU/X U, where PU is the distributed
probability of the type-U atom around the central type-V atom in the solid solution alloy and X U
is the molar fraction of the type-U atom. When α = 0, the type-U and type-V atoms have adisorderly distribution. Otherwise, in the case of α < 0, the type-U and type-V atoms areheterogeneous neighbors and are segregated neighbors in the case of α > 0.(24)The distribution of atoms in the two nearest shells for an Fe–Ni alloy can be calculated byapplying the Cowley parameters. The typical sequence of procedures and results provided inRef. 25 are summarized.
(1) For the Fe46.5Ni53.5 (at.%) alloy with the fcc single phase, the two sets of Cowley parameters
are α110 = −0.07665; α200 = 0.06463 and α110 = −0.1396; α200 = 0.10073, which are obtained byneutron diffraction experiments and Monte Carlo simulations.(25)(2) If the Ni atom is taken to be the central atom V, the mole fraction of U atoms (Fe atoms) is
XU = 46.5%; then, the probability of Fe atoms being around the Ni atom is found to be
P U = (1 − α)XU = (1 + 0.07665) * 46.5% = 50% by substituting α110 = −0.07665. That is, thecomposition of the 12 nearest shell atoms is Fe6Ni6.
(3) By replacing XU = 46.5% with α200 = 0.06463, the probability of an Fe atom appearing in thesecond shell, which is centered on the Ni atom, is PU = (1 − α)X U = (1 − 0.06463) * 46.5% = 43.5%.
That is, the composition is Fe0.87Ni1.13 for the six atoms in the second shell. It is concluded
that the nearest-neighbor shell atoms with Ni atom at the center are arranged as Ni–Fe6Ni6 –Fe0.87Ni1.13, where only two effective atoms in the second shell participate in the electron
cloud resonance of the central atom.
(4) Optimizing the formula of Ni–Fe 6 Ni6 –Fe 0.87 Ni1.13 using the integer ratio of atoms,
Ni–Fe7Ni5–Ni2 is derived with the Ni atom at the center. Otherwise, if the Fe atom is taken as
the center, the distribution of the nearest two-shell atoms becomes Fe–Ni8Fe4 –Fe2. The
chemical composition of the alloy is 53.3 at.% Ni, which is basically consistent with thecomposition of the alloy, and the deviation is −0.2%. This means that the composition of the
Fe–Ni alloy can be resolved by applying the model of atoms in the two nearest shells. The
feasibility of this model has also been verified in another Fe–Ni alloy.

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The feasibility of the aforementioned procedures has also been verified for various
components for Fe–Ni alloys in previous work.(26–30) The evaluated results are shown in Table 2.
As seen in Table 2, when the range of Ni contents is between 30.2 and 77.5 at.%, the 12 atoms in
the first shell centered on Ni are distributed between Fe8.58Ni3.42 and Fe3.01Ni8.93 according to theresults of evaluations using the Cowley parameters. Ideally, the atoms in the second shell and thecentral atom should be of the same type, Ni.
By adjusting all the atoms in the second shell as Ni atoms, the ideal CCF of all alloys is
obtained. The deviation of the Ni component ratio from the alloy component is not more than
0.3%. However, in the ideal CCF of the shell, the drawback is that the alloy cannot be resolved
using the integer formula, except for the alloy with 53.5 at.% Ni.
To mitigate the aforementioned drawback of the ideal CCF, each Fe–Ni alloy listed in Table 2
is further analyzed utilizing the double cluster model. The following key points are revealed inthis work:(1) When the content of Ni is 30.2 at.%, there are two types of nearest-neighbor shell cluster,
namely, [Ni–Ni8Fe4] with the central Ni atom and [Fe–Fe12] centered on an Fe atom. The twoclusters originate from the ordered phase Fe3Ni and the pure Fe atoms, respectively.
Otherwise, the cluster form [Ni–Ni8Fe4]Fe2 in this component does not satisfy the second
nearest-neighbor order owing to the Fe-rich composition. There are not enough Ni atoms toform the second nearest-neighbor order with the central atom.
(2) When the content of Ni is 36.8 at.%, there are two types of nearest-neighbor shell cluster,[Ni–Fe8Ni4] and [Fe–Fe8Ni4], which originate from the ordered phases FeNi and FeNi3. The
second-shell atoms form the second nearest-neighbor order with the central Ni and Fe atoms,
respectively. The double clusters are [Ni–Fe8Ni4]Ni2 and [Fe–Fe8Ni4]Fe2, respectively.
(3) When the content of Ni is 75–77.5 at.%, there are two types of nearest-neighbor shell cluster,[Fe–Ni8Fe4] and [Ni–Ni12], which belong to the ordered phase FeNi and the pure Ni atoms.
The second-shell atoms form the second nearest-neighbor order with the central Fe and Ni
atoms, respectively. The double clusters are [Fe–Ni8Fe4]Fe2 and [Ni–Ni12]Ni2, respectively.
According to the above discussion, it is concluded that the double cluster model with an
integer ratio of Fe and Ni atoms is suitable for analyzing all of the real Fe–Ni alloys shown in
Table 2. Furthermore, the analyzed results are also consistent with the results of calculations
using the Cowley parameters of the various Fe–Ni alloys. In the double cluster model, the firstnearest-neighbor atom arrangement is derived from three types of relatively stable ordered phasewith the pure Fe, pure Ni, and Fe–Ni alloys. The second nearest neighbor and the central atom
are basically of the same type, which satisfies the ideal atomic distribution. This is further proof
that all of the Fe–Ni alloys can be resolved by the double cluster model with two atoms in thesecond nearest-neighbor shell. That is, 15 atoms in the range of the nearest-neighbor atoms are
participating in the resonance.
In the fcc atomic lattice structure, there are 6 atoms in the second nearest neighbor. However,
the resonance of the first nearest-neighbor cluster with stronger relative force can only drive two
atoms in the second nearest neighbor to participate in the resonance. This is deduced from thespherical period theory as follows.

77.5(27)77.5(29)76.5(29)75(27)

75(30)

53.5(27)53.5(25)

36.8

(27)

36.8(28)

30.2

(27)

30.2(26)

Composition Ni
at.%Cowleyparametersα110 = –0.0201
α200 = 0.0451
α110 = –0.0282
α200 = 0.0562
α110 = –0.0583
α200 = 0.0522
α110 = –0.0797
α200 = 0.0675
α110 = –0.07665
α200 = 0.06463
α110 = –0.1396
α200 = 0.10073
α110 = –0.111
α200 = 0.136
α110 = –0.1305
α200 = 0.1388
α110 = –0.0868
α200 = 0.1005
α110 = –0.10821
α200 = 0.11948
α110 = –0.13664
α200 = 0.12729

Ni–Fe3.01Ni8.93 –Fe0.39Ni1.61

Ni–Fe3Ni9 –Fe0.4Ni1.6

Ni–Fe3.06Ni8.94 –Fe0.42Ni1.58

Ni–Fe3.39Ni8.61–Fe0.43Ni1.57

Ni–Fe3.33Ni8.67–Fe0.43Ni1.57

Ni–Fe6.36 Ni5.64 –Fe0.84Ni1.16

Ni–Fe6 Ni6 –Fe0.87 Ni1.13

Ni–Fe8.19Ni3.81–Fe1.18Ni0.82

Ni–Fe8.03Ni3.97–Fe1.20Ni0.80

Ni–Fe8.58Ni3.42–Fe1.31Ni0.69

Ni–Fe8.51Ni3.49 –Fe1.33Ni0.67

Two nearest-neighbor CCF
centered on Ni

Table 2Cluster formulas of Fe–Ni alloys.

Fe–Ni3Fe9 –Ni0.4Fe1.6

Fe–Ni3.06Fe8.94 –Ni0.42Fe1.58

Fe–Ni3.39Fe8.61–Ni0.43Fe1.57

Fe–Ni3.33Fe8.67–Ni0.43Fe1.57

Fe–Ni6.36Fe5.64 –Ni0.84Fe1.16

Fe–Ni6Fe6 –Ni0.87Fe1.13

Fe–Ni8.19Fe3.81–Ni1.18Fe0.82

Fe–Ni8.03Fe3.97–Ni1.20Fe0.80

Fe–Ni8.58Fe3.42–Ni1.31Fe0.69

Fe–Ni8.51Fe3.49 –Ni1.33Fe0.67

Two nearest-neighbor CCF
centered on Fe[Ni–Fe3.4Ni8.6]Ni2[Ni–Fe3.5Ni8.5]Ni2[Ni–Ni8.3Fe3.7]Ni2[Ni–Fe7 Ni5]Ni2[Ni–Fe9.5Ni2.5]Ni2[Ni–Fe10.5Ni1.5]Ni2

Ideal CCF centered on Ni
atoms[Fe–Fe0.4Ni11.6]Fe2[Fe–Fe0.5Ni11.5]Fe2[Fe–Fe0.7 Ni11.3]Fe2[Fe–Ni8Fe4]Fe2[Fe–Fe6.5Ni5.5]Fe2[Fe–Fe7.5Ni4.5]Fe2

Ideal CCF centered on Fe
atoms[Fe–Ni8Fe4]Fe2 +[Ni–Ni12]Ni2—[Ni–Fe8Ni4]Ni2 +[Fe–Fe8Ni4]Fe2[Ni–Ni8Fe4]Fe2 +[Fe–Fe12]Fe2

Bi-cluster formula

76.9

53.3

36.6

30

0.1

0.2

0.3

–0.2

–0.2

–0.2

Composition for biDeviation at.%cluster formula

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Sensors and Materials, Vol. 35, No. 10 (2023)

3

Electronic Model Analysis of Fe–Ni Alloys with Constant Expansion

3.1

Spherical periodic electronic resonant range in fcc solid solution alloys

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When the short range of fcc solid solution alloys participating in the Friedel spherical periodic
resonance is determined, the numbers of first nearest-neighbor clusters and second nearestneighbor atoms that are packed between these clusters within the resonant range are evaluated.
In this section, the resonant range of stable short-range ordered structures is obtained. Then, thechemical structure formula of the model is concluded to be [X–Y12]Z2, where X, Y, and Z are thecomposited atoms in the alloy.
There is an inheritance of the short-range ordered structure among liquid metals, amorphousalloys, and the ordered solid solutions.(31) On the basis of the principle of structural stability
corresponding to the lowest energy, whether in amorphous or solid solution alloys, thedistribution of electrons is consistent.
Häussler verified that when the maximum atomic distribution corresponds to the minimum
effective pair potential energy in molten or amorphous alloys, the energy possessed by theenergy band structure is minimal.(32) This causes the entire system to have the lowest energy andthe most stable structure.
When there are defects such as impurities or solute atoms in the bulk alloy, to obtain electric
neutrality, the electrons are rearranged to shield the bulk alloy from the effect of impurity
charges. The distribution of electronic density centered on the impurities exhibits the form of
long-range attenuation resonance. This resonance only exists within the range of a few atoms.
The oscillation of the electronic density around the ionic atom, which is an impurity atom asstated above, is called Friedel oscillation (FO). This oscillation forces the electrons to be locatedat different concentric spheres centered on the ion atom.(32)
FO is a result of the resonance of electrons and atoms in the gap position with the lowest
energy and the most stable structure. For the fcc solid solution alloys, the key point is todetermine the range or number of atoms involved in FO.
The effective pair potential energy ϕeff (r) for the electrons at distance r from the central ion
atom is(32)φeff ( r ) ∝ cos ( 2k F r + Θ ) / r 3 ,(1)

where kF is the Fermi radius of the distribution of electrons in the wave vector space and Θ is thephase offset. Experiments show that the phase offset is Θ = π/2 in the liquid metals andamorphous alloys with short- or medium-range orders.(32)
In real space, the band structure energy Ubs is given as

U bs ∝ ∫  g ( r ) − 1 φeff ( r ) r 2 dr ,(2)

when the effective pair potential energy ϕeff (r) of the electrons is at the local minimum, i.e.,
when the atomic density g(r) is at the local minimum, Ubs is minimum. This indicates that the4624

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ion is bound by FO at the local minima of ϕeff (r). The stable phase is formed when metal phasetransition occurs.(32)The position of the local minimum for ϕeff (r) is obtained asrn =1, 2, ,(1 / 4 + n ) λ Fr , n =(3)where λFr = 2π/2kF is the Friedel wavelength(32) and rn is the position of the nth-shell neighboratoms around the central ion atom.
Compared with the structure of liquid alloys, the solid solutions have the same stablestructure as FO for the local-range atoms. Truncated octahedral clusters (TOCs) are located not
only at the points of the sphere determined by FO but also at the points of the lattice determinedby the fcc structure, as shown in Fig. 1.
Figure 1 shows the distribution of the nearest two shell atoms. The TOC is formed by the firstnearest atoms, and the yellow atoms are the second nearest atoms of the central atom of the
TOC. If the central atom of one TOC, which is the big red atom in Fig. 1, is located at the origin,then its shell atoms are located at a distance of r1 = 1.25λFr from the origin. The central atoms of
the TOCs adjacent to the TOC at the origin, which are the big blue atoms in Fig. 1, are located at
a distance of r3 = 3.25λFr, and their nearest atoms are located at a distance of r 2 = 2.25λFr or fromthe origin.
The red curve for the effective pair potential energy of the electrons, ϕeff (r), is also shown in
Fig. 1. It tends to zero with increasing distance rn(λFr). This means that the short-range orderedstructure determined from FO is reduced to zero when r exceeds r4 = 4.25λFr.

Fig. 1.

(Color online) Spherical periodic short-range ordered structure.

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In three-dimensional space, a TOC has a strong interaction with its adjacent TOC. Electronicresonance is achieved through momentum exchange between the electrons and the ions within
the range of FO.(32)
When one layer of atoms slides relative to its neighbor layer of atoms, the slip is significantly
resisted by the FO structure. That is, the FO structure is sufficiently stable to prevent further
sliding. Therefore, the resistance to plastic deformation increases the strength as well as thehardness of the alloys.
There are a few packed atoms (PAs) between adjacent TOCs. The number of PAs, n, within
the FO range depends on the number of clusters in this range and the size of this range. The
structure of the TOCs and PAs is expressed by the chemical formula [X–Y12]–Zn, where [X–Y12]
is the cluster formed by the central atom of the nearest shell. X is the central atom of the TOC,
Y12 are the 12 nearest atoms in the structure of a truncated octahedron, and Zn (n is an integer)
are the second shell Z atoms.
3.2

First and second nearest-neighbor atomic models in fcc solid solution alloys

In fcc solid solutions, the key point in determining the chemical formula is the location of the
TOCs, which are located at the positions determined by the FO and fcc lattice structures.
In this study, one position for TOCs in the fcc alloy is introduced in Fig. 2. In Fig. 2(a), the fcc
lattice structure, lattice constant a, and TOCs are depicted. In Fig. 2(b), the parallelepiped, one

Fig. 2.

(a)

(b)

(c)

(d)

(Color online) Structural models of clusters for fcc alloys.

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TOC’s unit cell formed by eight TOCs, which is called the unicell of a TOC (UCTOC), is shown.
Its CSF is [X–Y12]Z2. The ratio of the number of TOCs to that of PAs is 1:2. In Fig. 2(c), a
midrange structure through outward growth from the unicell is called the super unicell of a TOC(SUCTOC) and has the quadrilateral axis symmetry. Figure 2(d) shows the structure in Fig. 2(c)from another perspective. It has the hexagonal axis symmetry.
In Fig. 2(a), for the fcc alloy, the distance between the shell atoms on the TOC and the origin
is r1 = 1.25λFr. Moreover,=
r1 a 2 ≈ 0.707 a, where a is the fcc lattice constant. The central
atom of the TOC closest to the origin is located at r3, which, in accordance with Eq. (3), isevaluated as
13 r
13 a
1

≈ 1.838a.
r3 = + 3  λFr = × 1 = ×
4 1.25 5
2
4


(4)

Eighteen different types of unicell model are evaluated for the fcc solid solution using Matlab.(33)
Their CSF is expressed as [X–Y12] –Zn, where n is the number of PAs.
When n = 2, that is, the TOC:PA ratio is 2, the model is called 2–2 and is shown in Fig. 2(b).
 1
 1
One TOC is at O(0,0,0) and there are three TOCS at A =( −3, − 1, 0 ), B = ( −2, 2, − 2 ), and
2
2
 1
C
=
( 0, − 1, − 3). The parallelepiped unicell is composed of these four TOCs as the basic vector,
2
as shown in Fig. 2(b). The polyhedral structure is obtained by growing the unicell outward in aplanar periodic sequence, as shown in Fig. 2(c). There are four TOCs in the first nearest shell.
Their central point P is at a distance of 1.581a from the central TOC at point M. There are eight
TOCs in the second nearest shell. Then, their central point Q is at a distance of 1.8701a frompoint M.
The average distance between the central point P or Q of the 12 TOCs and point M is obtained
as
=
R

1 n
=
Rk
∑
n k =1(1.8701a × 8 + 1.581a × 4=
) 12 1.774a,(5)

where Rk is the position of the kth cluster and n is the total number of clusters. Then, thedeviation between R̅ and the third position of FO, r3 ≈ 1.838a, is Δ = |r3 − R̅ | = 0.064a. The
lattice constant of iron is a = 2.86 × 10−10 min an iron-based alloy. Then, the deviation isΔ = 1.83 × 10−11 m. The ratio of the deviation Δ to the radius of the iron atom, rFe = 1.241 × 10−10,
is Δ/rFe = 0.147.
There are three layers of atoms within the distance of r3, so the deviation Δ is the difference
shared by three atoms. The average deviated distance of each atom from their original position is
0.07rFe. This is close to the amplitude of the thermal vibration of the atom on the lattice position,
indicating that the atoms do not truly deviate from the original lattice positions but merely
undergo thermal motion at their original positions. The 2–2 model provides the exact range forr3 in the fcc solid solution in accordance with the theorical value r3 = 3.25λFr determined using
the FO structure. The model also has high symmetry, as shown in Figs. 2(c) and 2(d).

Sensors and Materials, Vol. 35, No. 10 (2023)

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The deviation Δ and volume density ρ for the 18 different types of model reported in Ref. 33
are evaluated by the aforementioned calculation procedure and shown in Fig. 3. The volume
density is defined as

ρ = ( Number of atoms in the cut shell ) rcut3 ,(6)

where rcut = r3 is the radius of the cut shell, as shown in Fig. 1. In Fig. 3, it is observed that there
are five suitable candidate models, that is, 2–2, 3–1, 4–2, 5–5, and 6–3, with relatively small
deviation Δ = −0.0639, 0.0086, −0.0531, −0.0398, −0.0398 and high volume density ρ = 26.57,
21.99, 24.74, 21.38, 22.9 values, respectively. This means that the thermal vibrations of the atoms
within the FO structure all occur near the lattice positions in these five atomic models.
Because metal bonds have no directivity or saturation, the atoms in fcc alloys tend to bearranged with higher bulk density ρ values. The atomic volume density is higher and thepotential energy of the alloy is lower. They lead to a more stable alloy structure. The 2–2 modelhas the highest bulk density ρ and a petty deviation Δ among the 18 models reported in Ref. 33,
as shown in Fig. 3. This means that the structure of the 2–2 SUCTOCs has the excellent stability
to meet the conditions of both FO and the highest bulk density. The CSF of the structure isexpressed as [X–Y12] –Z2, where 2 is the number of PAs. This formula can be verified by the
explanation of Fe–Ni alloy grades in the next section.

4

Electronic Model Analysis of Fe–Ni-based Constant-expansion Alloy andDiscussion

Because their coefficient of thermal expansion matches that of many sealing materials, Fe–Ni
alloys are often used as sealing materials. The Ni content of these alloys ranges from 42 to 54
wt.%. The common characteristics of these alloys are good plasticity, electrical and thermal

Fig. 3.

(Color online) Deviation Δ and volume density ρ values of atomic models.

4628

Sensors and Materials, Vol. 35, No. 10 (2023)

conductivity, and processability. In addition, the key point is that these materials remain in theoriginal γ phase during storage and working.(34) They do not undergo phase transition, whichmeans that their stability is extremely high.
In the following, the 2–2 model with the analysis steps introduced in Sect. 3 is applied toanalyze the Fe–Ni-based constant-expansion alloys. The results are listed in Table 3, where thewt.% is the abbreviation for the weight percentage of Ni in Fe–Ni alloys. The Fe–Ni alloys listed
in Table 3 can be analyzed using the formula [X–Y12]Z2 of the 2–2 model. The chemicalcompositions of the elements in the formula are basically consistent with those specified by the
manufacturer. From the results in Table 3, the following is concluded.
(1) The composition of 42–54 Ni wt.% Fe–Ni sealing alloys was analyzed using [X–Y12]Z2,
where Y is the combination of Fe and Ni atoms and Z is the Fe or Ni atom. The total numberof atoms was 15. The composition formula for these binary alloys was in the range of Ni6Fe9
to Ni8Fe7.
(I) Two components were analyzed using the single cluster model. The CSFs were [Ni–
Fe8Ni4]Ni2 (47.9 wt.%) and [Fe–Ni8Fe4]Fe2 (54.5 wt.%).
(II) Three components were analyzed using the double cluster model. The CSFs were [Fe–Ni8Fe4]Fe2 + [Fe–Fe8Ni4]Fe2 (41.2 wt.%), [Fe–Ni8Fe4]Fe2 + [Ni–Ni4Fe8]Fe2 (44.6 wt.%),and [Fe–Ni8Fe4]Fe2 + [Ni–Fe8Ni4]Ni2(51.3 wt.%).

Table 3Cluster formulas and properties of Fe–Ni alloys.
Specification

Range ofspecification
Ni wt.%*

Chemical formula

Component
formula for 15
atoms

Ni wt.% inchemicalformulaNi42

41.5–42.5[Fe–Ni6Fe6]Fe2

Fe9Ni641.2

Ni43

42.5–43.5[Fe–Ni6.27Fe5.73]Fe2

Fe8.73Ni6.27

43

Ni45

44.5–45.5[Fe–Ni6.57Fe5.43]Fe2

Fe8.43Ni6.57

45

Ni48
Ni50
Ni52
Ni54

47–48
49.5–50.5
51.5–52.5
53.5–54.5
41.5–42.5
5.5–6.3
rem
46–48
5–6
rem
28.5–29.5
16.8–17.8
rem
32.5–34
13.6–14.8
rem
28.5–29.5
19.5–20.5
rem[Ni–Fe8Ni4]Ni 2[Fe–Ni7.31Fe4.69]Fe2[Fe–Ni7.61Fe4.39]Fe2[Fe–Ni8Fe4]Fe2

Fe8Ni7Fe7.69Ni7.31
Fe7.39Ni7.61
Fe7 Ni8[Cr–Ni6Fe6]Fe2—[Ni–(Fe7Cr)Ni4]Ni 2—[Co–Fe8.5(Co1.5Ni2)]Ni2—[Ni–Fe8Ni4]Co2—[Co–Fe8Ni4]Co2

—

47.9
50
52
54.5
Ni: 41.4
Cr: 6.1
Fe: remNi: 48.1
Cr: 6.1
Fe: remNi: 27.4
Co: 17.2
Fe: remNi: 34.2
Co: 13.7
Fe: remNi: 27.4
Co: 20.6
Fe: rem

Ni42Cr6

Ni47Cr5

Ni29Co18

Ni33Co14

Ni29Co20

Ni wt.% inBi-cluster chemical formula bicluster chemical
formula[Fe–Ni8Fe4]Fe2
41.2[Fe–Fe8Ni4]Fe2[Fe–Ni8Fe4]Fe2
42.9[Fe–Fe8Ni4]Fe1.5Ni0.5[Fe–Ni8Fe4]Fe2
44.6[Ni–Ni4Fe8]Fe2—[Fe–Ni8Fe4]Fe2
51.3[Ni–Fe8Ni4]Ni 2—
Ni: 41.4[Fe–Fe8Ni4]Cr2
Cr: 6.1[Fe–Ni8Fe4]Fe2
Fe: rem—Ni: 27.4
Co: 17.2
Fe: rem[Fe–Fe8Ni4]Ni 2[Co–Co4Fe8]Ni 2——

Sensors and Materials, Vol. 35, No. 10 (2023)

4629(III) There was another component that could not be resolved into an integer ratio cluster
expression. Its two-cluster CSFs were [Fe–Ni8Fe4]Fe2 + [Fe–Fe8Ni4]Fe1.5Ni0.5.
(IV)There were three types of TOC in the first neighbor clusters for all Fe–Ni sealing
alloys, namely, [Fe–Ni8Fe4], [Ni–Fe8Ni4], and [Fe–Fe8Ni4], which were derived fromthe ordered phases of FeNi and FeNi3, as shown in Table 1.
(V) The number of atoms in the second shell of the structural formulas was 2 in all cases. It
was proven that the 2–2 model has the highest compactness of the fcc solid solution.
(2) For the analysis of the ternary Fe–Ni sealing alloys, the results are summarized for thenickel–chromium–iron alloys with Ni42Cr6 and Ni47Cr5 in sequence.
(I) The CSFs were [Fe–Fe8Ni4]Cr2 + [Fe–Ni8Fe4]Fe2 and [Ni–(Fe7Cr)Ni4]Ni2.
(II) The double-cluster CSF of the Ni42Cr6 alloy, [Fe–Fe8Ni4]Cr2 + [Fe–Ni8Fe4]Fe2, wasobtained by replacing the second shell atom in the first CSF of [Fe–Fe8Ni4]Fe2 + [Fe–Ni8Fe4]Fe2 with a Cr atom in the first CSF of [Fe–Fe8Ni4]Fe2.
(III) Similarly, the CSF of the Ni47Cr5 alloy, [Ni–(Fe7Cr)Ni4]Ni2, was obtained by replacing
an Fe atom in the nearest-neighbor shell of [Ni–Fe8Ni4]Ni2 in the CSF of the Ni–48
alloy with a Cr atom.
(IV)The above results are consistent with the mixing enthalpy of the alloys, wherein
ΔHFe–Ni = −2 KJ/mol, ΔHCr–Ni = −7 KJ/mol, and ΔHFe–Cr = −1 KJ/mol.(35) The mixing
enthalpy reflects the dispersion and mixing degree of atoms in the system, and theabsolute values of the mixing enthalpy of Fe and Cr atoms are the smallest. This meansthat these two atoms have the least force and can be substituted.
(3) The results are summarized for the iron–nickel–cobalt alloys with Ni33Co14, Ni29Co20, andNi29Co18 as follows.
(I) The CSFs of such alloys were [Ni–Fe8Ni4]Co2, [Co–Fe8Ni4]Co2, and [Fe–Fe8Ni4]Ni2 +[Co–Fe8]Co4Ni2, respectively.
(II) The CSFs of the Ni33Co14 and Ni29Co20 alloys were obtained by replacing the Ni atomwith the Co atom in the CSF of the Ni48 alloy, [Ni–Fe8Ni4]Ni2.
(III) The CSF of the Ni29Co18 alloy was obtained by replacing five Ni atoms with five Co
atoms in the double-cluster CSF of [Fe–Fe8Ni4]Ni2 + [Co–Fe8]Co4Ni2.
(IV)The above CSFs are consistent with the mixing enthalpy of the alloys,
wherein ΔHFe–Ni = −2 KJ/mol, ΔHCo–Ni = −0 KJ/mol, and ΔHFe–Co = −1 KJ/mol.(35)
Because the mixing enthalpy of Co and Ni atoms is zero, the two atoms areinterchangeable in structural formulas of the iron–nickel–cobalt alloys.
In the aforementioned structural formulas of the Fe–Ni sealing alloys, the atoms in the firstshell are different from the central atom, and the atoms in the second shell are of the same typeas the central atom.
The structural formula of the 2–2 model for the nearest two-shell atoms, which is obtained
from the spherical periodic ordered model, is basically consistent with the relationship between
the attractive and repulsive forces determined by the mixing enthalpy of atoms. In Table 3, theanalyzed components of common Fe–Ni sealing alloys basically fall into the composition rangesof the alloys specified by the manufacturer. Only a few values for the alloys are slightly off,
which are underlined. It is further proved that the 2–2 model developed from the spherical4630

Sensors and Materials, Vol. 35, No. 10 (2023)

periodic ordered model, which is described in Sect. 3, can be used to analyze all the Fe–Ni
sealing alloys.
Because there is no long-range order in the disordered solid solution alloy, the short-rangeordered structure only exists in a small range of FO. The heterogeneous clusters different fromthe TOCs, such as holes, dislocations, and interstitial sites, prevent the growth of the short-rangeordered structure. Therefore, it is not easy to form the long-range ordered chemical structures ofwhole crystals.
The fcc alloy compositions obtained by similar procedures have been reported.(36–39) For
example, the Cu–Ni–Si ternary alloys were analyzed by selecting the nearest-neighbor clusters
from the precipitated and strengthened phases.(36,37) The number of second nearest-neighboratoms was directly assigned the value of 1 or 3 without proposing any assignment principles.
The binary copper alloy was analyzed by selecting the nearest-neighbor cluster from Cowleyparameters and determining the number of second nearest-neighbor atoms from the empiricalvalues of amorphous alloys and the density of the densest stacked crystal structure.(38,39) In thisstudy, the determination and selection of the number of nearest second-shell atoms are based on
the ordered phases, Cowley parameters, the maximum bulk density of atoms, and FO. Comparedwith the previously obtained results,(36–39) the proposed methods can enhance the reliability of
the analysis of alloy compositions.

5.

Conclusions

In the study, the tools for analyzing metal alloys, such as Cowley parameters, FO, and the
NNTS atomic model, were addressed. They were also applied to analyze Fe–Ni binary or eventernary sealing alloys successfully. The conclusions of the work are shown below.
(1) The chemical composition formula of the NNTS model was inferred from [X–Y12]Zn where n
is an integer. X–Y12 is composed of the first nearest–neighbor shell atoms. The combinations
Ni–Ni8Fe4, Fe–Ni12, Ni–Fe8Ni4, Fe–Ni8Fe4, Ni–Fe12, and Fe–Fe8Ni4 are included in X–Y12.
(2) Various Fe–Ni alloys with nickel contents in the range of 30 to 80 at.% were analyzed using
one or two types of [X–Y12]Z2 model. The results were consistent with those calculated using
Cowley parameters.
(3) Compared with the most stable structure of the alloy, it was pointed out that the 2–2 model isthe most stable structure in the fcc solid solution. The atomic density of the 2–2 model is thehighest in the short range of the Friedel spherical periodic resonance structure.
(4) Both Fe–Ni binary and ternary alloys were analyzed using two types of [X–Y12]Z2 model. Our
results revealed an effective approach for the composition design of alloys.

Acknowledgments
This work was supported in part by Projects for the Department of Science and Technologyof Fujian Province (Grant nos. 2020-H-0049, 2021-H-0060, and 2021-G-02013), the Operational
Funding of the Advanced Talents for Scientific Research (Grant nos. 19YG04 and 19YG05) ofSanming University, and the Natural Science Foundation of Sanming University (Grant no. 113/Sensors and Materials, Vol. 35, No. 10 (2023)

4631

A160002). The authors also acknowledge the support from the School of Mechanical andElectric Engineering, Sanming University.

References
1 P. Edinger, G. Jo, C. P. Van Nguyen, A. Y. Takabayashi, C. Errando-Herranz, C. Antony, and K. B. Gylfason:
Opt. Express 31 (2023) 6540. https://doi.org/10.1364/OE.480219
2 S. S. K. Ardestani, V. Dashtizad, and A. Kaflou: Ceram. Int. 47 (2021) 2008. https://doi.org/10.1016/j.
ceramint.2020.09.032
3 M. Jing, G. Ni, C. Zhu, Z. Li, G. Wang, Z. Wang, and Q. Huang: Constr. Build. Mater. 377 (2023) 131076.
https://doi.org/10.1016/j.conbuildmat.2023.131076
4 P. Yu, F. Zhan, W. Rao, Y. Zhao, Z. Fang, Z. Tu, Z. Li, D. Guo, and X. Wei: Micromachines. 14 (2023) 84.
https://doi.org/10.3390/mi14010084
5 M. Wei, F. He, X. Cao, B. Zhang, C. Zheng, and J. Xie: Ceram. Int. 48 (2022) 27718. https://doi.org/10.1016/j.
ceramint.2022.06.072
6 M. M. Uddin, J. Jie, C. Wang ,C. Zhang, and W. Ke: Energy Build. 286 (2023) 112939. https://doi.org/10.1016/j.
enbuild.2023.112939
7 Q. Sun, G. Sun, Y. Liu, H. Chen, M. Li, C. Huang, and W. Zhu: J. Mater. Sci-Mater. El. 32 (2021) 16230. https://doi.org/10.1007/s10854-021-06171-3
8 D. M. Chen, L. F. Hu, Y. Z. Xue, S. Chen, and W. Wang: Trans. China Weld. Inst. 39 (2018) 71. https://doi.
org/10.12073/j.hjxb.2018390227
9 Q. Lu,W. Li, X. Zhang, Z. Liu, Q. Cao, X. Xie, and S. Yan: Fuel 263 (2020) 116690. https://doi.org/10.1016/j.
fuel.2019.116690
10 H. L. Hong, C. H. Yang, K. C. Wang, and H. X. Yan: Mod. Phys. Lett. B 35 (2021) 2141009. https://doi.
org/10.1142/S0217984921410098
11 I. Y. Khairani, Q. Lin, J. Landers, S. Salamon, C. D. Buendia, E. Karapetrova, H. Wendi, G. Zangari, and B.
Gökce: Nanomaterials 13 (2023) 227. https://doi.org/10.3390/nano13020227
12 Y. Wu, F. Zhang, X. Yan, H. Huang, X. Wen, Y. Wang, M. Zhang, H. Wu, X. Liu, and H. Wang: J. Mater. Sci.
Technol. 62 (2021) 214. https://doi.org/10.1016/j.jmst.2020.06.018
13 F. D. C. Garcia Filho, R. O. Ritchie, M. A. Meyers, and S. N. Monteiro: J. Mater. Res. Technol. 17 (2022) 1868.
https://doi.org/10.1016/j.jmrt.2022.01.118
14 S. Bao, X. W. Guo, Z. L. Wang, Q. Zhao, Y. L. Liu, F. S. Meng, B. Z. Sun, N. Jia, and Y. Qi: Scripta Mater. 224(2023) 115132. https://doi.org/10.1016/j.scriptamat.2022.115132
15 Y. Liu, J. Luo, and X. Huang: J. Alloy. Compd. 932 (2023) 167591. https://doi.org/10.1016/j.jallcom.2022.167591
16 Y. Li, D. Yang, and W. Qiang: J. Alloy. Compd. 931 (2023): 167588. https://doi.org/10.1016/j.jallcom.2022.167588
17 D. He, G. Liu, J. Liu, H. Xie, D. Yue, Z. Chen, C. Wei, X. Xu, and H. Xie: Corros. Sci. 211 (2023) 110917.
https://doi.org/10.1016/j.corsci.2022.110917
18 A. Abu-Odeh, and M. Asta: Acta Mater. 226 (2022) 117615. https://doi.org/10.1016/j.actamat.2021.117615
19 R. Ma, J. Rong, S. Zhang, H. Wang, and G. Yun: J. Phys. Soc. Jpn. 89 (2020) 044705. https://doi.org/10.7566/
JPSJ.89.044705
20 A. Toriyabe, W. T. Chiu, A. Umise, M. Tahara, K. Goto, H. Kanetaka, T. Hanawa, and H. Hosoda:
Intermetallics. 139 (2021) 107349. https://doi.org/10.1016/j.intermet.2021.107349
21 S. Jiang, F. Zhou, W. Liu, S. Liu, X. Wang, S. Zhang, X. Zhu, R. Wei, P. Zheng, and Z. Lin: Fusion Eng. Des.
193 (2023) 113658. https://doi.org/10.1016/j.fusengdes.2023.113658
22 D. D. Johnson, J. D. Althoff, F. J. Pinski, J. B. Staunton, and M. F. Ling: J. Phase Equilib. 18 (1997) 598. https://doi.org/10.1007/BF02665818
23 J. S. Wróbel, D. Nguyen-Manh, Y. M. Lavrentiev, M. Marek, and S. L. Dudarev: Phys. Rev. B, 91 (2015)
024108. https://doi.org/10.1103/PhysRevB.91.024108
24 D Gehringer, M Friák, and D Holec: Comput. Phys. Commun. 286 (2023) 108664. https://doi.org/10.1016/j.
cpc.2023.108664
25 X. Jiang, G. E. Ice, C. J. Sparks, L. Robertson, and P. Zschack: Phys. Rev. B. Condens Matter. 4 (1996) 3211.
https://doi.org/10.1103/PhysRevB.54.3211
26 P. Cenedese, F. Bley, and S. Lefebvre: Mater. Sci. 21 (1983) 351. https://doi.org/10.1557/PROC-21-351
27 S. M. Bokoch, V. A. Tatarenko, and I. V. Vernyhora: Usp. Fiz. Met. 13 (2012) 269. http://dspace.nbuv.gov.ua/
handle/123456789/98336
28 J. L. Robertson, G. E. Ice, C. J. Sparks, X. Jiang, P. Zschack, F. Bley, S. Lefebvre, and M. Bessiere: Phys. Rev.
Lett. 82 (1999) 2911. https://doi.org/10.1103/PhysRevLett.82.2911

4632

Sensors and Materials, Vol. 35, No. 10 (2023)

29 G. E. Ice, C. J. Sparks, A. Habenschuss, and L. B. Shaffe: Phys. Rev. Lett. 68 (1992) 863. https://doi.org/10.1103/
PhysRevLett.68.863
30 S. Lefebvre, F. Bley, M. Fayard, and M. Roth: Acta Metall. 29 (1981) 749. https://doi.org/10.1016/00016160(81)90118-8
31 L. Zhang, S. Su, W. Fu, J. Sun, Z. Ning, A. H. W. Ngan, and Y. Huang: Compos. Part B: Eng. 258 (2023)
110698. https://doi.org/10.1016/j.compositesb.2023.110698
32 P. Häussler: J. Phys. Colloques. 46 (1985) 361. https://doi.org/10.1051/jphyscol:1985854
33 H. L. Hong, C. Dong, and Y. Zhang: Acta Phys. Sin. 65 (2016) 036101. https://wulixb.iphy.ac.cn/article/
doi/10.7498/aps.65.036101
34 W. Liu, C. Chen, Y. Tang, Q. Long, S. Wei, G. Zhang, F. Mao, Q. Jiang, T. Zhang, and M. Liu: Calphad. 69(2020) 101763. https://doi.org/10.1016/j.calphad.2020.101763
35 A. Takeuchi, and A. Inoue: Mater. Trans. 46 (2005) 2817. https://doi.org/10.2320/matertrans.46.2817
36 D. M. Li, J. Y. Han, and C. Dong: Acta Phys. Sin. 68 (2019) 196102. https://doi.org/10.7498/aps.68.20190593
37 D. M. Li, Q. Wang, B. B. Jiang, X. X. Li, W. L. Zhou, C. Dong, H. Wang, and Q. X. Chen: Prog. Nat. Sci.:
Mater. Int. 27 (2017) 467. https://doi.org/10.1016/j.pnsc.2017.06.006
38 Z. Li, D. D. Dong, L. Zhang, S. Zhang, Q. Wang and C. Dong: Sci. Rep. 12 (2022) 3169. https://doi.org/10.1038/
s41598-022-06893-2
39 C. Dong, D. D. Dong, and Q. Wang: Acta Metall. Sin. 54 (2018) 293. https://www.ams.org.cn/CN/10.11900/041
2.1961.2017.00462