review h-r paper revised

8/12/2019 Review H-R Paper Revised

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The prediction of solid solubility of alloys: developments and applications of

Hume-Rotherys rules

Y.M. Zhang

!. R.". #vans$

%. Yanga

1. Centre for Materials Research and School of Engineering and Materials Science

Queen Mary, University of London

Mile End Road

London E1 4NS

. !e"art#ent of Che#istry,

University College London,

$ %ordon Street,

London &C1' $(), U*

a(uthor to +ho# corres"ondence should e addressed. E#ail- S.yang/#ul.ac.u0,

1
mailto:[email protected]:[email protected]


2/47

&bstract:

n the 12$s, 'u#e3Rothery hel"ed to #a0e the art of #etallurgy into a science y

the discovery of rules for the "rediction of soluility in alloys. heir si#"licity and

generality #ade the# eco#e one of the #ost i#"ortant rules in #aterials science.n

the fe+ decades after 'u#e3Rothery5s discovery, #any researchers have tried to

#a0e 6corrections7 to '3R rules ai#ing to #a0e the# +or0 etter in general alloy

syste#s. hose researches included e8"lanations of the rules using /uantu# and

electron theories and ne+ co#inations of the factors to give etter #a""ing. n this

"a"er, +e revie+ #ost of these contriutions and introduce recent "rogress in

soluility "rediction using artificial neural net+or0s.

*ey+ords- soluility, alloys, 'u#e3Rothery rules, electronegativity, artificial neural

net+or0.


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.' (ntroduction

he #ost +idely acce"ted vie+ of scientific #ethod is ased on the creative

e#ergence of hy"otheses or con9ectures :1; +hich gradually eco#e +ell3trenched in

the for# of estalished theories as #ore su""orting e8"eri#ental evidence is sought

and found. ( contrasting #odel for scientific discovery is attriuted to 1?@131@@A in +hich large a#ounts of data are first collected, asse#led into tales,

surveyed and fro# +hich theories are devised :;. ( central deate in the history and

"hiloso"hy of science focuses on these contrasting e8"lanations of scientific #ethod

and is concisely articulated y %illies in his analysis of artificial intelligence :B;.

'u#e3Rothery5s rules, occu"ying a central s"ace at the heart of #etallurgy, are in the

=aconian tradition. n the 12$5s, after surveying the availale soluility data, 'u#e3

Rothery distinguished the factors that influence co#"ound for#ation and control

alloying ehaviour.here e8ists a connection et+een soluility, ato#ic sie, crystal

structure and a "articular concentration of valence electrons in an alloy :4, ?;. 'u#e3

Rothery added other ideas, develo"ing conce"ts +hich are no+ 0no+n collectively as

'u#e3Rothery5s rules :@3D;.


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Miutani :1G;, Massals0i :4, 1D; and as descried in iogra"hical s0etches aout

'u#e3Rothery :12;. Researches on '3R rules losso#ed in the 12B$s312D$s ut the

"rediction of soluility +as gradually su"erseded y calculation of "hase diagra#s

>C(LH'(!A.'o+ever, the si#"licity and generality of '3R rules still #a0e the#

one of the #ost i#"ortant cornerstones in #aterials science. &atson and &einert in

$$$ :$; #entioned #ost of these rules are still useful today as in 'u#e3Rothery5s

ti#e, and discussed the a""lications to the transition and nole alloys, oth +ith each

other and +ith #ain grou" ele#ents. HarthI :1; discussed a""lication of D3N and

valence electron rules for the intl "hases >+hich aredefined as 6se#i#etallic or even

#etallic co#"ounds +here the underlying ionic3covalent onds "lay such an

i#"ortant role that che#ically ased valence rules can e used to account for

stoichio#etry and oserved structural features7A and their e8tension. Recently these

rules have also een used in nanocrystal gro+th and co#"ound for#ing tendency :;

as +ell as ther#odyna#ically staility of ordered structures in FJ se#iconductor

alloys :B;. n this revie+, the authors ai# to address so#e recent "rogress on the

develo"#ent and a""lication of '3R rules.

(s cautioned y Hettifor :12;, ecause different rules +ere e8"ressed or stressed y

'u#e3Rothery at different ti#es, it is so#eti#es difficult to define +hat constitutes

K'u#e3Rothery Rules5 and this confusion is e8tant. here is general agree#ent that,

in order of i#"ortance, the ato#ic sie factor is first, follo+ed y the electro3

negativity effect. he i#"ortance of the electron concentration >eFa ratioA in

deter#ining solid soluility oundaries is recognised in so#e cases ut other factors

are rarely discussed in sufficient detail. Surveying #etallurgical and "hysical science

4


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"ulications in general, different sources e8"ress 'u#e3Rothery5s ideas using ter#s

such as- effects, "rinci"les, factors or "ara#eters :4;.

$.' #arly )ormation and Revision of Hume-Rotherys Rules

n 12?, after ta0ing his Hh! under Sir 'arold Car"enter at the Royal School of

Mines, 'u#e3Rothery returned to 8ford and +or0ed on inter#etallic co#"ounds,

#etallogra"hy and che#istry, e8tending his ideas on the for#ation of co#"ounds.

'e e8a#ined "hase diagra#s of the nole and related #etals >i.e. Cu, (g and (uA,

es"ecially those alloyed +ith the =3sugrou" ele#ents >these include Li, =e, =, C, N,

,


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n the 12B$s, 'u#e3Rothery shifted his attention to characterisation of ato#ic sie y

nearest neighour distance instead of volu#e :@;. +o of the 'u#e3Rothery rules

controlling solid soluility +ere discovered- 1A the first 'u#e3Rothery rule, the

ato#ic sie factor, said that if the ato#ic dia#eters of the solvent and solute differ y

#ore than aout 1431?O then the "ri#ary solid soluility +ill e very restricted A

the second rule e#"hasised the i#"ortance of the electron concentration >or electron

"er ato# ratioA in deter#ining the "hase oundary. =oth of these rules are "resented

in his classical "a"er in 12B4 :@;.

(lthough 'u#e3Rothery at that ti#e had found t+o i#"ortant guidelines +hich

decided the for#ation of "ri#ary solid solutions, he +as unclear ho+ to classify

inter#etallic co#"ounds. n 12BG, after studied the silver rich anti#ony3silver alloy

syste# +ith Reynolds :G;, 'u#e3Rothery eca#e a+are of a third factor restricting

solid soluility, that is, electroche#ical factor #a8i#u# solid soluility reduced as

the electronegativity difference et+een solute and solvent increased ecause of the

co#"etition to for# inter#etallic co#"ounds.

he relative valence rule +as #entioned in the 12B4 "a"er, and the i#"ortance of this

rule +as stressed y 'u#e3Rothery in early editions of his fa#ous oo0 The

Structure of Metals and Alloys:G;. 'o+ever, in his later versions of this oo0, it is

stated K#ore detailed e8a#ination has not confir#ed this and, in its general for#, the

su""osed "rinci"le #ust no+ e discarded5 :D, 2;.

*.' )urther development and application of Hume-Rotherys Rules

@


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he develo"#ent of 'u#e3Rothery5s Rules can e classified into t+o categories- he

first is the develo"#ent +ithin each rule, and the second is the develo"#ent of

ana#or"hoses or alternatives of these rules as a +hole in order to get #ore "o+erful

and "recise "redictions. n the first category, researchers, 1A "rovided e8"lanations of

s"ecific rule>sA fro# ele#entary electron theory, A "ointed out the +ea0ness and

deficiency of individual rules. n the second category, researchers have atte#"ted to

e8tend the rule>sA or itsFtheir alternatives to +ider a""lications. n +hat follo+s, the

discussion "rogresses along these t+o "aths.

*. +evelopment and application of each rule

*.. &tomic %i,e )actor

he ato#ic sie factor rule is usually "resented in the follo+ing +ay :@;- 6if the

ato#ic dia#eters of the solute and solvent differ y #ore than 14O, the soluility is

li0ely to e restricted ecause the lattice distortion is too great for sustitutional

soluility.7 &hen the sie factor is unfavourale, the "ri#ary solid soluility +ill e

restricted +hen the sie factor is favourale, other factors li#it the e8tent of solid

soluility and it is of secondary i#"ortance. &aer et al.:B$; a""lied the sie factor

alone to 14B ter#inal solid solutions and +ithin 2$.B1O >??2F@12A of the syste#s

+here lo+ solid soluility +as "redicted, lo+ solid soluility +as indeed oserved.

n the other hand, it +as less easy to "redict extensivesolid soluility +hen there +as

a s#all sie difference it achieved only a ?$ O >4$BFD$4A success rate. he sie factor

rule has een e8"lained y using ele#entary electron theory. t can e sho+n that :B13

B4;, if a #isfitting solute ato# is regarded as an elastic s"here +hich is then

co#"ressed or e8"anded into a hole of the +rong sie in the solute lattice, +hich can

G


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e treated as an isotro"ic elastic continuu#, the ensuing total strain energy E in oth

the #atri8 and solute can e esti#ated as

B

$D rE= , >E/uation 1A

+here P is the shear #odulus and r$and >1Ar$are the unstrained radii of the solvent

and solute ato#s. a0ing as $.14 >as the sie factor rule declaresA or $.1?, andB

$r

$.G eJ, this gives TkE B4 at 1$$$ *. !ar0en and %urry :1$; "roved that at

te#"erature , the "ri#ary solid soluility +ould e restricted to elo+ aout 1 at.O

+hen the energy of solution e8ceeds 40= "er ato#, +here 0= is =olt#ann5s

constant. (lthough elastic theory cannot e a""lied strictly at the ato#ic level, this

gives a si#"le e8"lanation of 'u#e3Rothery5s sie factor rule.

Mott :BB; "rovided a /uantu# #echanical asis for elastic theory ased the &igner3

Seit +ave function T$>rA, for an electron in the lo+est state in a &igner3Seit cell.

he +avefunction for an electron in the alloy can e e8"ressed a""ro8i#ately as T>rA

u >rA e8" >ik rA, +ith u >rA having the for#s of an ( ato# or a = ato# &igner3Seit

+avefunction T$>rA inside the &igner3Seit cells of ( or = ato#s >if ( and = are of

the sa#e valencyA in the alloy, +here ris the vectorial "osition of the electron and kis

the +ave3vector.


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he validity of the sie factor has een deated since the rule +as "ro"osed. 'u#e3

Rothery et al. the#selves "ointed out :@; that the e8act 6ato#ic dia#eter7 of an

ele#ent is al+ays difficult to define. hey defined the ato#ic dia#eter as given y

the nearest3neighour distance in a crystal of the "ure #etal. 'o+ever, this dia#eter

cannot necessarily e transferred to the alloy ecause 1A the Kradius5 of an ato# is

"roaly affected y coordination nu#er. E8ce"t for the heavy ele#ents, ele#ents

of the = su3grou"s tend to crystallie +ith coordination nu#er D3N, +here N is the

grou" to +hich the ele#ent elongs. his is due to the "artly covalent nature of the

forces in these crystals and, e8ce"t in %rou" J =, results in the ato#s having t+o

sets of neighours at different distances in the crystal. A n so#e structures there are

great variations in the closest distance et+een "airs of ato#s at their closest distance

of a""roach, de"ending on the "osition and direction in the lattice. BA n for#ing a

solid solution, the Ksies5 of individual ato#s #ay change according to the nature and

degree of local dis"lace#ents. n the case of anisotro"ic or co#"le8 structures or

+here the coordination nu#ers are lo+, the closest distance of a""roach does not

ade/uately e8"ress the sie of the ato# +hen in solid solution :1D;.


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he ato#ic di#ensions can e calculated y using "seudo"otential theory, such as the

+or0 done y 'ayes et al. :B@; on Li3Mg, nglesfield :BG3B2; on 'g, Cd and Mg

alloys, 'ayes and Woung :4$; on al0ali alloys, Stroud and (shcroft :41; for Cu3(l,

Li3Mg and Cu3n, Meyer et al.:4344; on analying the diffusion ther#o"o+ers of

dilute al0ali #etal alloys, on calculating the lattice s"acings and co#"ressiilities of

non3transition ele#ent solids and for analying residual resistivities in silver and gold

and Singh and Woung :4?; on heats of solution at infinite dilution. hey can also e

otained fro# the free3electron #odel develo"ed y =roo0s :4@; and have een used

y Magnaterra and Meetti :4G, 4D;.

he actual individual ato#ic sies can also e esti#ated fro# static distortions in a

solid solution y #odulation in diffuse X3ray scattering :42, ?$; or fro# +ea0ening of

the interference #a8i#a analogous to ther#al effects :?13?4;.


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investigation has not confir#ed the generality of this rule. (n e8a#"le is that

#onovalent silver can dissolve aout $O alu#iniu# ut trivalent alu#iniu#

dissolves aout 4O silver. 'o+ever, for high valence, covalently onded

co#"onents, the relative valence factor a""lies +ell. E/uation A

11


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+here I is the ioniation energy, A is electron affinity, and Y is Mulli0an

electronegativity. !ividing y .D, gives a""ro8i#ately Hauling5s e#"irical scale.

&atson and =ennett :@$; "oint out that in the case of transition #etals, the "artly

filled d states at energies near the


13/47

n addition to these efforts to e8"lain each rule using /uantu# or electronic theory,

another category of develo"#ent has een to #a" the soluleFinsolule syste#s in a

t+o di#ensional diagra#.

*.$ Mapping and derivatives of Hume-Rotherys Rules

(lthough e8tensive researches have een done to +or0 out the theory ehind the '3R

rules, it +ould e very useful if the soluility of the #aterials can e #a""ed

diagra##atically on a Cartesian syste#. So researchers can si#"ly calculate the

coordinate of the ele#ent to "redict the soluility using such a diagra#. his

direction started fro# !ar0en and %urry :1$;, follo+ed y %schneidner :?G;,

Cheli0o+s0y :11;, (lonso et al.:1, 1B;, and hang :14, 1?; a#ong others.

*.$. +aren-"urry method

n 12?B, !ar0en and %urry :1$; "ro"osed a diagra##atic #ethod to descrie the

solid soluility of fifty alloy syste#s >!% #ethodA. hey used the sie factor as

ascissa and electroche#ical factor >electronegativityA as ordinate to "lot soluilities

of each alloy syste# and then dra+ an elli"se >rsolventZ 1?O as #inor a8is, YsolventZ $.4

as "rolate a8is, +here rsolvent and Ysolvent are radius and electronegativity of solvent

res"ectivelyA to se"arate the solule ele#ents fro# the insolule ele#ents. he result

is sho+n in


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unsuccessful classification in silver syste#s +as, at that ti#e, attriuted to the use of

una""roved electronegativity values of silver ut %schneidner :?G; argued that even if

a""roved electronegativity values for silver +ere used, the classification +ould still e

"oor.

&aer et al.:B$; e8a#ined the universality of the 'u#e3Rothery sie rule and the

!% #ethod for "redicting solid soluilities. (fter analyed 14?? inary alloy

syste#s, they confir#ed 'u#e3Rothery5s sie factor and sho+ed that the

electronegativity is an i#"ortant consideration in the for#ation of solid solutions.

>insert figure 1 hereA

n 12D$, %schneidner :?G; again a""lied the !% #ethod to create a soluleFinsolule

classification y introducing the effect of crystal structure >electronic3crystal structure

!ar0en3%urry #ethod, ECS!%A.


15/47

e8"ected 4A. n cases +here the solvent is ansp3ele#ent and solute is a d3ele#ent, no

solid solutions are "redicted to for#, regardless of either the crystal structure or sie

factors. (""lying the ECS!% rules to ten solvents >Mg, (l, called electronic and crystal structure, sie #odel, ECSA, and

"ointed out that their #ethod should also e /uite good at "redicting the e8tent of

another ele#ent in a inary co#"ound. n co#"ounds, iA the co#"atiility of the

crystal structure of the solute +ith either or oth of the co#"onents of the

inter#etallic "hase is regarded as a critical factor iiA the valence of the solute

co#"ared +ith the co#"onents is also decisive iiiA if the aove t+o criteria are

favourale, then ato#ic sie factor +ould e the final decisive issue and ecause of

the less elastic nature of co#"ounds co#"ared +ith ele#ental #etals, a Z1$O

li#itation on ato#ic sie should e a""lied.

*.$.$ /helio0sys method

n the 12G$s, t+o events occurred that led :11; to the vie+ that so#e of the arriers to

understanding solid soluility in inter#etallic alloys could e re#oved. ne +as that

Miede#a and collaorators "redicted and classified heats of for#ation for regular

inter#etallic alloys +hich is "redo#inantly deter#ined y the electronegativity

difference > \ A and the difference in electron density at the oundary of the

&igner3Seit cell > WSn A of "ure #etals :@43@D;. he other +as that *auf#ann and

co3+or0ers develo"ed ion3i#"lantation techni/ues and conducted ion3i#"lantation to

1?


16/47

"rovide a +ide range of ne+ and uni/ue #etastale alloy syste#s, unotainale y

conventional #etallurgical "rocedures :@23G1; +hich led to suse/uent efforts to

study solid soluility in alloys :G3G?;.

n 12G2, Cheli0o+s0y introduced a gra"hical "rocedure si#ilar to the !ar0en3%urry

#ethod to analyse solid soluility in the case of divalent hosts. n his +or0, a different

"air of coordinates +ere introduced- the electron density at the oundary of ul0

ato#ic cells, WSn , and the electronegativity\ . (s #entioned efore, these t+o

coordinates are the funda#ental "ara#eters in a successful se#i3e#"irical theory of

heats of alloy for#ation develo"ed y Miede#a and co3+or0ers :@43@D;. n his ne+

0ind of "lot, Cheli0o+s0y +as ale to give #ore reliale "redictions. (n e8a#"le of

Cheli0o+s0y "lots and a co#"arison +ith !ar0en3%urry "lots are sho+n in insert figure hereA


17/47

Hauling5s scale, +hile in Cheli0o+s0y "lots the Miede#a scale is used. =oth scales

actually have a good correlation as sho+n y Miede#a et al. :@4;. he other co3

ordinate is different, ato#ic sie is used in !ar0en3%urry "lots and electron cell3

oundary density is used in Cheli0o+s0y5s #ethod. (lthough higher accuracy has

een estalished, so#e e8ce"tions re#ain. hese e8ce"tions suggest that

Cheli0o+s0y5s #ethod is still susce"tile to so#e i#"rove#ent :1;.

*.$.* &lonsos Method

n the 12D$s, fro# analysis of oth !ar0en3%urry and Cheli0o+s0y #ethods, (lonso

and Si#oar :1; "ro"osed a sche#e containing all three coordinates >ato#ic sie,

electronegativity, electron cell3oundary densityA. he suggestion +as also "ro"osed

y Miede#a and !e Chatel :2B;. =y incor"orating a sie factor in a ne+ gra"hical

#ethod, they i#"roved on the original Miede#a coordinate sche#e "ro"osed y

Cheli0o+s0y. n their analysis, each che#ical ele#ent is characteried y three

"ara#eters- the ato#ic volu#e V , the electronegativity \ , and the electron

density at the oundary of ul0 ato#ic cells bn . (s the result,\ and sn >the

difference of electronegativity and electron density at the oundary of ul0 ato#ic

cellsA co#ined into a ne+ "ara#eter, !" , the heat of for#ation of an e/ui3ato#ic

co#"ound, calculated using a se#i3e#"irical for#ula "ro"osed y Miede#a,

( ) ( ) #n$%" s! +=

BF1

\ , >E/uation BA

+here H and Q are universal constants and R is another constant +hich deviates fro#

ero only +hen one of the #etals is "olyvalent +ithpelectrons :@?, @D, 24;. hen, the

t+o "ara#eters !" and V >e8"ressed as &igner3Seit radius, R&A are used to

construct a t+o3di#ensional #a". n this #a", the che#ical ele#ents insolule in a

given host are neatly se"arated fro# the solule ones y a straight line. he e8a#"les

1G


18/47

and the co#"arisons +ith Cheli0o+s0y5s "lot are sho+n in aA and B>A. n

these "a"er, the relative i#"ortance of the three coordinates >ato#ic sie,

electronegativity and electron cell3oundary densityA is de#onstrated also they

e8"lained the success of the sche#es of !ar0en and %urry and of Cheli0o+s0y y the

fact that ato#ic sie and electron cell3oundary density are strongly correlated in

given class of #etals. Later, (lonso et al.a""lied this #ethod to the "rediction of

solid soluility in nole #etal, transition #etal and sp#etal ased alloys :1B, 2?;.

)ones has a""lied this #ethod to the solid soluility of t+o light #etals, #agnesiu#

and alu#iniu# :2@;.

(fter the heat of for#ation calculations y Miede#a, others succeeded in "redicting

the heats of for#ation of different inary alloys fro# oth first3"rinci"les and se#i3

e#"irical #ethods :2G31$D;. !uring the last t+enty years and ased on calculated

heats of for#ation of alloys and se#i3e#"irical theories +ith "ara#eters such as

electronegativity difference, ato#ic dia#eter and nu#er of covalent onds, a large

nu#er of "redicted #a8i#u# solid soluilities of alloys, design of alloy syste#s or

for#ation of co#"ounds have a""eared :1$2312;. his indicates these #ethods still

have vitality in the "rediction of alloy for#ation even though near thirty years has

"assed.

>insert figure B hereA

*.$.1 Zhang 23 Method

hang and co3+or0ers used gra"hical #ethods +ith various "ara#etersFcoordinates to

"redict the for#ation of a#or"hous alloys and solid solutions. Several factors affect

the for#ation of a#or"hous alloys and solid solutions, so#e acting against each other

>e/uation GA. :1?, 1B$;.

1D


19/47

n 12DB :1B1;, they a""lied Miede#a5s coordinates to the "rediction of inary

a#or"hous alloy for#ation and found that this #ethod +or0ed /uite +ell. More

s"ecifically, they found >1A the ranges of for#ale and non3for#ale inary

a#or"hous alloys can e, to a good e8tent, se"arated y a straight line, +hich is

$11

B21$BF1

\ =

sn , >E/uation 4A

+here \ and sn are the differences of electronegativity and electron density

res"ectivly at the oundary of ul0 ato#ic cells as #entioned efore >A of the D

a#or"hous alloys for#ed y #elt3/uenching, the "rediction accuracy +as 2.GO >BA

of the ?4 non3for#ale a#or"hous alloys, the "rediction accuracy +as @@.GO >4A of

the 1 a#or"hous alloys for#ed y non3#elt /uenching, the "rediction accuracy +as

4G.@O. f the total 1?G alloys studied, the overall accuracy +as GG.GO. hese results

co#"ared +ell +ith the +or0 done y Shi et al. :1B; using ond3"ara#etric

diagra#s, in +hich the ond "ara#eters &r' F and !(Vr' F >+here the ' is ato#ic

valence, &r is ato#ic 0ernel radius e/uals a""ro8i#ately the "ositive ionic radius

not including the valence electrons, and !(Vr is covalent radiusA +ere used to "redict

the for#ation of inary a#or"hous alloys, and the "rediction accuracy +as D$O.

a0ing the "ara#eters used in Miede#a5s coordinates +ithout the sie factor, hang

:1BB; co#ined the t+o che#ical coordinates \ andBF1

sn into one-

( ) 1F1B21$ BF1\ = sny , >E/uation ?A

and using the radius difference of ele#ents

12


20/47

O

1

1

#

##x

= >E/uation @A

as the other coordinate to constructed a t+o3di#ensional #a". insert tale 1 hereA

hang and co3+or0ers "ro"osed another gra"hical #ethod, +hich co#ined &r' F

and !(Vr' F as an electron factor, then together +ith a sie factor to "rovide t+o

coordinates for the study of solid soluility, they descried the for#ation of

a#or"hous alloys :1B4;. he t+o coordinates are e8"ressed y >1A ond3"ara#etric

function %%&

xr

'x

r

'

+B

B

1

cov

and >A half of the e#"irical interato#ic distance

#res"ectively. &here &r' F is the ratio of ato#ic valence and ato#ic 0ernel radius,

and !(Vr' F the ratio of ato#ic valence and covalent radius. he *ernal radius

$


21/47

e/uals a""ro8i#ately the "ositive ionic radius. %x is the Hauling electronegativity.

&hen the coordinate "oint of a host ele#ent is re"resented in the chart, the closer is

the "oint of a solute ele#ent to it, the s#aller the differences of the electron factors

and of the sie factors et+een the solute ele#ent and the host ele#ent. hang :1B$;

a""lied this #ethod to 1$D$ inary and got a "rediction accuracy of 2BO.

hang used a #odified electron factor

( )%B%AA!(V

%A

B&

))

r

')

r

'y

+

=

B

B

1 >E/uation GA

as ordinate and O1

1

#

##x

= as ascissa to search for solid soluilities at roo#

te#"erature in 4@$ inary alloys. n this +or0 the transition #etals of the fourth,

fifth and si8th long "eriods and 1D non3transition #etals are studied. E/uation DA

1


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Hredictions of solid soluility of other alloy syste#s using first "rinci"les can e

found else+here :14?31@@;.

1.' %olubility prediction using artificial neural net0ors

(lthough a large nu#er of researches on soluility "rediction have een "ulished in

the last century, fe+ of the# have atte#"ted to "redict the soluility /uantitatively.

he sie factor has een regarded as the #ost i#"ortant factor ut has never een

/uantitatively "roved. Recently, the authors :1@G; have used one tool of artificial

intelligence, artificial neural net+or0s >(NNsA, to si#ulate the "rocess that 'u#e3

Rothery used to derive the 'u#e3Rothery5s Rules fro# e8"eri#ents. 1A ato#ic sie "ara#eter, >A

valence "ara#eter, >BA electroche#ical "ara#eter, i.e. electronegativity and >4A

structure "ara#eter of solvent and solute ato#s. hree different e8"ressions of these

"ara#eters +ere used to e8a#ine +hich gave the est "erfor#ance-

1. he ra+ data that 'u#e3Rothery used.

. he original collected values for each "ara#eter of solvent and solute ato#s.

B. he functionalied "ara#eters-

B


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iA -or the sie factor. he difference et+een the ato#ic dia#eters of solvent

and solute ato#s divided y the dia#eter of the solvent ato#s.

iiA -or the valence factor. riginal values are used, leaving the neural

net+or0 to decide the relations et+een valence of solvents and solutes.

iiiA -or the electroche,ical factor. he difference et+een that of the solvent

and solute ato#s.

ivA -or the structure para,eter. he structures are e8"ressed in three sets of

nu#ers re"resenting "ri#itive cell di#ensions, angles and syste#s. he

three sets are >1A unit cell length, >A a8es angles and >BA >si#"le ase3

centred face3centred ody3centredA.

&hen the original values of in"ut "ara#eters are used, the training "erfor#ance of the

(NN is /uite good >+ith regression coefficient R$.22@A ut the "rediction of the

testing set is "oor. &hen the functionalied values +ere used as in"ut "ara#eters,

oth the training and testing set of the (NN gave good "erfor#ance. hrough a

search of all co#inations of those "ara#eter for#ats, the est for#at of in"ut

"ara#eters +as deter#ined as the functionalied values.

(nother uncertainty in the '3R rule is the ato#ic sie difference- so#e researchers

elieve the threshold is 14O and others elieve it5s 1?O. he "erfor#ance of (NN

using the 1?O criterion is slightly higher than that for 14O +hich i#"lies 1?O is a

etter threshold in the sie factor.

Using the sa#e a""roach and including the 1?O criterion, the structure "ara#eter +as

introduced in ter#s of +hether the structure of solvents and solutes are the sa#e or

4


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not >i.e. 1 sa#e, $ not sa#eA. here +as no i#"rove#ent in correlation. his indicates

that the structure "ara#eter do not "lay a very i#"ortant role in soluility.

1.$ +etermination of the output parameters

at.OA fro# e8"eri#ent results and y3a8is are

"redicted soluility >at.OA "redicted fro# (NN. Most of the data "oints are located

close to y8 indicating an accurate "rediction.

1.* +etermination of the relative importance of each rule

t has een recognied that each rule has a different influence on soluility- the #ost

i#"ortant rule is the ato#ic sie factor, follo+ed ne8t y the electroche#ical factor

>electronegativityA. 'o+ever, these "ara#eters are not +holly inde"endent of each

other their inter"lay #a0es the deter#ination of soluility very difficult. So the

deter#ination of the relative i#"ortance of each rule is not easy. n this research, the

"erfor#ance of the (NN +as evaluated +hen so#e of the rules +ere delierately

o#itted. Using the #ean error of the testing set as the #ain criterion for accuracy of

?


26/47

the "rediction, ato#ic sie has the strongest effect ecause, +hen it is o#itted, the

error is highest. Electronegativity has a stronger influence than valence. &hen "airs of

"ara#eters are o#itted, the "erfor#ance of the "rediction eca#e +orse. #ission of

the structure "ara#eter only had s#all effect on the "erfor#ance of the (NN, +hich

i#"lies that the structure "ara#eter does not "lay a very i#"ortant role, and indeed

'u#e3Rothery did not include it in 12B4.

6.' %ummary

!uring the decades after 'u#e3Rothery5s Rules +ere "ulished, the value and

reliaility of the# has een discussed e8tensively. hey have een deated, e8tended

and refined in different +ays, as #entioned in the te8t. (lso, as 'u#e3Rothery and

co3+or0ers stated at the ti#e +hen they first raised these rules- Kn general, the

soluility li#it is #ainly deter#ined y these factors, and it is their inter"lay that

#a0es the results so co#"le85 :@;. 'o+ever, even though one cannot "redict the solid

soluility li#its accurately using 'u#e3Rothery5s rules, they are still useful

guidelines for 9udging the soluility of alloy syste#s or for#ation of inter#etallic

co#"ounds.

Massals0i :4; suggests that although 'u#e3Rothery5s rules are i#"ortant, insufficient

docu#ented e8a#"les of "ractical a""lication and "redictive ca"aility have een

de#onstrated so far and this is a real challenge to the future of alloy develo"#ent and

of inter#etallic co#"ound for#ation. (t "resent, it is necessary to accu#ulate a large

a#ount of +or0 to investigate +hich rules have een deter#ined and +hich have not.

Ne+ investigation tools, such as ( >(rtificial ntelligenceA, can e used to validate

those rules and to e8"lore ne+ "ara#eters or rules in the future.

@


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&cno0ledgments

he authors are grateful to School of Engineering and Materials Science of Queen

Mary, University of London to su""ort this +or0 y "roviding research studentshi" to

W.M. hang.

G


28/47

References

:1; *. R. Ho""er. Con9ectures and Refutations- he %ro+th of Scientific

*no+ledge. Routledge, (ingdon. U*. 12@B. "". BB3@?.

:;


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:1?; =. &. hang and S. . Liao. Hrogress on the theories of solid soluility of

alloys >"artA. Shang'ai Met. 1, 1222, B31$.

:1@; . =. Massals0i and '. &. *ing. (lloy "hases of the nole #etals. Hrog.

Mater Sci. 1$, 12@B, B3GD.

:1G; . =. Massals0i and U. Miutani. Electronic3structure of 'u#e3Rothery

"hases. Hrog. Mater. Sci. , 12GD, 1?13@.

:1D; . =. Massals0i. Structure and staility of alloys. n- R. &. Cahn and H.

'aasen eds. Hhysical Metallurgy. Ne+ Wor0- North 'olland, 122@. "". 1B?3

$4.

:12; !. %. Hettifor. &illia# 'u#e3Rothery- his life and science. n- E. (. urchi,

R. !. Shull and (. %onis eds. Science of alloys for the 1st century- a

'u#e3Rothery sy#"osiu# celeration. &arrendale- MS >he Minerals,

Metals ^ Materials SocietyA, $$$. "". 23B.

:$; R. E. &atson and M. &einert. he 'u#e3Rothery _"ara#eters_ and onding

in the 'u#e3Rothery and transition3#etal alloys. n- E. (. urchi, R. !. Shull

and (. %onis eds. Science of alloys for the 1st century- a 'u#e3Rothery

sy#"osiu# celeration. &arrendale- MS >he Minerals, Metals^ Materials

SocietyA, $$$. "". 1$?3112.

:1; E. Harthe.


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:2; &. 'u#e3Rothery, R. E. S#all#an and C. &. 'a+orth. he Structure of

Metals and (lloys. ?th ed. London- Metals and Metallurgy rust of the

nstitute of Metals and the nstitution of Metallurgists, 12@2.

:B$; ). . &aer, *. (. %schneidner, (. C. Larson and W. H. Margaret. Hrediction

of solid soluility in #etallic alloys. rans. Metal. Soc. (ME. G, 12@B,G1G3GB.

:B1; ).


31/47

:4?; S. Singh and &. '. Woung. 'eats of solution for si#"le inary alloys. ). Hhys.


32/47

:@1; &. =. Hearson. he Crystal Che#istry and Hhysics of Metals and (lloys. Ne+

Wor0 London- &iley3nterscience, 12G. ".@D.

:@; *. W. Li and !. ECSA #odel for "redicting inary solid solutions. Hrog. Mater. Sci. 42,

$$4, 41134D.

:@4; (. R. Miede#a. Electronegativity Hara#eter for ransition3Metals 3 'eat of


33/47


34/47

:2; X. C. %ui, S. . Liao, '. &. Xie and =. &. hang. Haraola #odel of

for#ation la+ of /uasicrystal ased on the fourth transition #etals. Rare Met.

Mater. and Eng. B?, $$@, 1$D$31$D4.

:2B; (. R. Miede#a and H.


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:1$@; %. =oolo, ).


36/47

:1$; L. *. eles, ).


37/47

:1BB; =. &. hang. !escri"tion of the for#ation of inary a#or"hous3alloys y

che#ical coordinates. Hhysica = ^ C. 1B, 12D?, B123B.

:1B4; =. &. hang. he effect of sie factor in the for#ation of a#or"hous alloys.

(ctaMetall Sin. 1G, 12D1, D?32.

:1B?; =. &. hang. Se#i3e#"irical theory of solid soluility in inary3alloys. .

Metal0d. G@, 12D?, @43G$.

:1B@; =. &. hang and S. . Liao. heory of solid soluility for rare earth #etal

ased alloys. . Hhys. =- Condens. Matter. 22, 122@, B?34B.

:1BG; R. &. lesins0i and %. ). (aschian. he Cu3Si >Co""er3SiliconA syste#. ).

Hhase E/uili. G, 12D@, 1G$31GD.

:1BD; (. unger, S. '. &ei, L. %.


38/47

:14D; S. Wa#a#oto, . &a0aayashi and '. *oayashi. Calculation of solid

soluility and co#"ound for#aility of (l3alloys y e8tended 'uc0el3#ethod.

). )"n. nst. Met. ?G, 122B, 1B@G31BGB.

:142; &. . Luo and M. E. Schlesinger. her#odyna#ics of the iron3caron3inc

syste#. Metall. Mater. rans. =. ?, 1224, ?@23?GD.

:1?$; =. 'alle#ans H. &ollants and ). R. Roos. her#odyna#ic reassess#ent and

calculation of the


39/47

:1@; M. '. 138ASi solid solution

structural staility. ). (lloys Co#"d. 4@?, $$D, 4@34@G.

:1@G; W. M. hang, S. Wang and ). R. %. Evans. Revisiting 'u#e3rothery`s rules

+ith artificial neural net+or0s. (cta Mater. ?@, $$D, 1$24311$?.

B2


40/47

ale Ca"tions-

ale 1 Co#"arison of the for#ation of inary a#or"hous alloys using

different coordinates >redra+n fro# :12;A

ale Co#"arison of the "rediction of solid soluilities y different #ethods

>redra+n fro# :1?;A

4$


41/47

ales-

ale 1

Hure Miede#a5s

coordinate criterion

Miede#a5s

coordinate "lus sie

factor

=ond3"ara#etric

diagra#

>1A (#or"hous alloys

y #elt /uenching

total D D B

+rong @ 1$ B

accuracy >OA 2.G DG.D 2$.@

>A Non3for#ale

a#or"hous alloys

total ?4 ?4 ?G

+rong 1D 2 G

accuracy >OA @@.G DB.B DG.G

>BA (#or"hous alloys

y non3#elt/uenching

total 1 1 BD

+rong 11 G 1$

accuracy >OA 4G.@ @@.G GB.G

otal accuracy >OA >1A >A D.4 D@ DD.D

>1A >A >BA GG.G DB.4 D4.B

41


42/47

ale

No. of alloy syste#s and the "rediction accuracy

otal No. Hrediction accuracy O

'u#e3Rothery5s rule >sie factor onlyA 14B @G.@

!3% Method 14?? G@.@Cheli0o+s0y Method 12 D

(lonso Method B4 2$

hang =& Method >"araola se"arationA BD@4 DG.

hang =& Method >elli"se se"arationA BD@4 2$.B

4


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>aA

>A


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>aA

>A


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0 20 40 60 80 100-20

0

20

40

60

80

100

Experimental Solubility T (at.%)

PredictedSolubilityA(at.%)

Best Linear Fit: A = (0.977) T + (0.23)

R = 0.993

0 20 40 60 80 100-20

0

20

40

60

80

100


PredictedSolubilityA(at.%)

Best Linear Fit: A = (0.962) T + (-1.19)

R = 0.992

0 20 40 60 80 100-20

0

20

40

60

80

100


PredictedSolub

ilityA(at.%)

Best Linear Fit: A = (0.966) T + (0.0162)

R = 0.992

Data Points

Best Linear Fit

A = T

(a) Training set (b) Testing set

(c) Whole set

review h-r paper revised

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