mitigating grain growth in binary nanocrystalline alloys through solute selection based on...

Computational Materials Science 84 (2014) 255–266

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Mitigating grain growth in binary nanocrystalline alloys through soluteselection based on thermodynamic stability maps

0927-0256/$ - see front matter Published by Elsevier B.V.http://dx.doi.org/10.1016/j.commatsci.2013.10.018

⇑ Corresponding authors.E-mail addresses: [email protected] (K.A. Darling), mark.tschopp@

gatech.edu (M.A. Tschopp).

K.A. Darling a,⇑, M.A. Tschopp a,*, B.K. VanLeeuwen b, M.A. Atwater c, Z.K. Liu b

a U.S. Army Research Laboratory, Weapons and Materials Research Directorate, Aberdeen Proving Ground, MD 21005, USAb Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA 16802, USAc Applied Engineering, Safety & Technology Department, Millersville University, Millersville, PA 17551, USA

a r t i c l e i n f o

Article history:Received 15 July 2013Received in revised form 30 August 2013Accepted 9 October 2013Available online 31 December 2013

Keywords:Nanocrystalline materialsGrain growthGrain boundary energyGrain boundary segregation

a b s t r a c t

Mitigating grain growth at high temperatures in binary nanocrystalline alloys is important for processingnanocrystalline alloy systems. The objective of this research is to develop a methodical design-basedapproach for selecting solutes in binary nanocrystalline alloys by revisiting grain boundary thermody-namics and the internal processes of grain growth and solute segregation in a closed system. In this work,the grain boundary energy is derived and systematically studied in terms of temperature, grain size, con-centration, and solute segregation for binary systems of 44 solvents and 52 solutes, using readily-avail-able elemental data, such as moduli and liquid enthalpy of mixing. It is shown that through solutesegregation, the grain boundary energies of some binary systems can be reduced, resulting in thermody-namically stable grain structures and successful prediction of solutes that inhibit grain growth in somenanocrystalline alloys. Parametric studies reveal trends between equilibrium grain size, solute distribu-tion and temperature for various binary systems culminating in the generation of nanocrystalline ther-modynamic stability maps as a tool for solute selection in binary nanocrystalline alloys.

Published by Elsevier B.V.

1. Introduction

Nanocrystalline (nc) alloys [1] are an important subset of metal-lic materials due to their small grain size (<100 nm) which impartsproperties and potential applications that may not be achievableusing conventional coarse-grained polycrystalline materials. Forinstance, they tend to possess extremely high strengths [2–8]that are associated with grain boundary strengthening (i.e., theHall–Petch effect [9,10]). A fundamental limitation to their usecomes from their inherent thermal instability, which has beenextensively explored using both experiments [11–19] and compu-tational approaches [20–33]. The small grain size produces anextremely large driving force for grain growth. If not kept in check,grain growth can occur at modest temperatures, even at room tem-perature in pure materials such as copper and palladium [34].Since bulk nanocrystalline alloys are often produced via mechani-cal alloying [35,36], this grain growth phenomenon provides a sig-nificant obstacle to consolidating the nanocrystalline powders bytraditional powder metallurgy techniques where high temperaturesintering is common.

There are several methods that have been devised to circum-vent this temperature sensitivity and stabilize the grain size.

Grain size stabilization commonly involves adding small quanti-ties of an insoluble element (i.e., solute). Nonequilibrium process-ing (e.g., mechanical alloying, rapid solidification, etc.) is oftenused to force the solute into solution and, upon heating, the sol-ute segregates to interfaces such as grain boundaries [37,38].Alternatively, the solute will remain in solution or precipitateout as a second phase. These two basic mechanisms for stabiliza-tion are known as thermodynamic and kinetic stabilization,respectively [39]. For kinetic stabilization, the solute acts to hin-der grain boundary mobility by diffusion-related means such assolute drag [40] or by pinning boundaries with a fine dispersionof precipitates [41]. For thermodynamic stabilization, solute is ex-pected to segregate to grain boundaries and reduce the grainboundary energy so as to minimize or eliminate the driving forcefor grain growth [16,20,32,42–49]. Since grain boundary energy isthe driving force for grain growth, a reduction in grain boundaryenergy can impede or even entirely inhibit grain growth. Thereduction in grain boundary energy provided by a segregatingsolute is determined by the segregation energy, DGseg

[16,20,32,42–49]. Since the values of DGseg are usually not avail-able, they are estimated. Wynblatt and Chatain [50] recently re-viewed the analytical models on segregation to grainboundaries (GBs) and surfaces and addressed the difficulty ofmeaningful definitions of segregation enthalpy, entropy, and freeenergy among various issues. The central equation for all modelsis as follows for a binary system:

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256 K.A. Darling et al. / Computational Materials Science 84 (2014) 255–266

xGBB

1� xGBB

¼ xIB

1� xIB

exp �DGex

seg

RT

" #; ð1Þ

where xGBB and xI

B are the mole fractions of component B (solute) inthe grain boundary and grain interior, respectively, DGex

seg is the ex-cess Gibbs energy of segregation, and R and T are the gas constantand absolute temperature, respectively. The Gibbs energy of segre-gation is thus defined as follows:

DGseg ¼ DHseg � TDSseg ¼ DHseg � TDSexseg

� �� TDSideal

seg

¼ DGexseg � TDSideal

seg ; ð2Þ

where DHseg and DSseg are the enthalpy and entropy of segregation,respectively, and the entropy of segregation can be broken into twoparts: the excess entropy of segregation DSex

seg and the ideal entropyof segregation DSideal

seg . The ideal entropy of segregation is defined asthe ideal entropy change of the system, including both grain andgrain boundary regions, due to segregation:

DSidealseg ¼ �R ln

xIAxGB

B

xIBxGB

A

: ð3Þ

where the component A refers to the solvent. Hence, Eq. (1) is ob-tained by substituting the functional form for the entropy term(Eq. (3)) and setting Eq. (2) equal to zero (i.e., DGseg ¼ 0), whichindicates zero driving force for solute segregation to the interface(e.g., see Ref. [50]). With a given model for DGex

seg the compositionsin the grain boundary and grain interior can be obtained and arefurther used to evaluate the grain boundary energy of isotropic oranisotropic systems.

Conventionally, the interfacial energy is defined as the revers-ible work needed to create a unit area of surface (e.g., grain bound-ary) at constant temperature, volume (or pressure), and chemicalpotentials, i.e., for an open system [50]. However, in practical appli-cations, the interfacial energy is measured in a closed system (i.e.,constant compositions). In the present work, we will first derivethe expression of grain boundary energy for a closed system by dif-ferentiating the internal and external variables and defining theinternal processes, and then predict the effect of segregating sol-utes on the grain boundary energy in binary systems of 44 solventsand 52 solutes. Here, we are only considering two internal pro-cesses: grain growth and grain boundary segregation. Thus, weare assuming that no secondary phases are forming, which is a rea-sonable approximation in the dilute case where phase formationwill be kinetically hindered [42]. Section 2 describes the thermody-namic details of the present model. This reduction in grain bound-ary energy modeled herein can then be used to select binarynanocrystalline alloys in terms of thermal stability maps. Section 3applies the present model first to the case of Fe–Zr and then con-siders a large number of binary systems in several relevant metals.The significance of this research is in the development of thermo-dynamic stability design maps as a way to organize and select alloycompositions for nanocrystalline grain growth mitigation.

2. Thermodynamics of grain growth and segregation

The combined first and second laws of thermodynamics of asystem can be written as [51]

dG ¼ �SdT þ VdP þX

i

lidNi � Ddn; ð4Þ

where G, S, V, and li are the Gibbs energy, entropy, volume, andchemical potential of component i of the system; T, P, and Ni aretemperature, pressure, and moles of component i controlled fromsurroundings; n and D represent the extent of an internal processand its driving force and more than one simultaneous internal pro-

cesses can be considered as shown below. For a system with a grainboundary area of A which reaches a metastable equilibrium underconstant temperature, pressure, and compositions, each term inEq. (4) becomes zero, i.e., dG ¼ 0.

Let us now consider an internal process with the grain bound-ary area changed by dA. Due to the composition difference betweenthe grain boundary and the grain interior, there will be a simulta-neous re-distribution of elements, commonly referred to as segre-gation. These two internal processes contribute to the change ofGibbs energy of the system as follows [51]:

dG ¼X

i

lGBi � lI

i

� �dni þ c0dA; ð5Þ

where lGBi and lI

i are the chemical potentials of component i in thegrain boundary and grain interior, respectively, dni is the change ofcomponent i from the grain interior to the grain boundary, and c0 isthe change in G due to a change in A at constant ni, i.e.,

c0 ¼@G@A

� �ni ;T;P;Ni

: ð6Þ

lGBi � lI

i can be further written as

lGBi � lI

i ¼ RT lnxGB

i

xIi

� �þ Gex=GB

i � Gex=Ii ; ð7Þ

where Gex=GBi and Gex=I

i are the partial excess Gibbs energy of compo-nent i in the grain boundary and grain interior, respectively. For abinary A–B system, we have dnA ¼ �dnB, and Eq. (5) can be re-writ-ten as

dG ¼ RT lnxGB

B

1� xGBB

1� xIB

xIB

� �þ DGex

seg

� dnB þ c0dA

¼ DGsegdnB þ c0dA; ð8Þ

where DGexseg and DGseg are defined as

DGexseg ¼ Gex=GB

B � Gex=IB � Gex=GB

A � Gex=IA

� �¼ Gex=GB

B � Gex=GBA � Gex=I

B � Gex=IA

� �; ð9Þ

and

DGseg ¼@G@nB

� �A;T;P;NA ;NB

; ð10Þ

It is evident from the above equations that the grain boundary en-ergy for a closed system is defined as [52]

c ¼ @G@A

� �T;P;NA ;NB

¼ DGseg@nB

@Aþ c0; ð11Þ

There are two significant observations from the above derivations:(1) DGex

seg is related to the partial quantities of excess Gibbs energy ofgrain and grain boundary, not the excess Gibbs energies themselves,and (2) c ¼ c0 when DGseg ¼ 0, meaning there is no redistribution ofelements during a change of grain boundary area. The latter is aconstrained equilibrium, because it poses a limitation on an internalprocess for a closed system, which would otherwise take place toreduce the Gibbs energy of the system. Therefore, the widely usedEq. (1), commonly referred to as Langmuir–McLean (or Fowler–Guggenheim) segregation isotherm is applicable when the redistri-bution of elements is negligible during measurements of grainboundary energies. If this is not the case, Eq. (11) should be usedfor grain boundary energy.

There are many models on how to treat DHseg and DSexseg in Eq. (2)

as discussed by Wynblatt and Chatain [50] including their depen-dence on five degrees of freedom of grain boundaries. In the pres-ent model, the grain boundary character, or the distribution ofgrain boundaries, is not explicitly modeled. Rather, it is anticipated

Fig. 1. Bilayer grain boundary used for the regular solution model. zl indicates in-plane bonds and zv indicates out-of-plane bonds. The latter can be bonds joiningatoms across the grain boundary (GB) or bonds joining atoms on the GB to the graininterior (GI). Solvent (A) and solute (B) atoms are shown on the GB bilayers.

K.A. Darling et al. / Computational Materials Science 84 (2014) 255–266 257

that as the grain size decreases, the grain boundary takes on ametastable structure and the impact between different boundarytypes decreases, in part due to the increasing role of triple junc-tions. However, other approaches involving broken bond modelsand bond interaction energy distributions have been utilized withvarying degrees of success [27,31,32,50]. As the purpose of thiswork is to compare the effects of a wide range of alloying elements,we follow the Wynblatt and Ku model for surface segregation [53]and assume DSex

seg ¼ 0. Therefore,

DHseg ¼ DHchemseg þ DHelastic

seg ; ð12Þ

where

DHchemseg ¼ cBrB � cArAð Þ 1� að Þ

� 2Xz

zl xGBB � xI

B

� �� zv xI

B �12

� �þ azv xGB

B �12

� �� ; ð13Þ

and

DHelasticseg ¼ �24pNAvKBGArArB rB � rAð Þ2

3KBrA þ 4GArB: ð14Þ

In the above equations, cA; cB; rA, and rB are the free surface ener-gies and the grain boundary areas per mole of grain boundary atomsof components A (solvent) and B (solute), respectively. a is a bondenergy interaction parameter introduced to distinguish the bondenergy in the lattice region from the grain boundary region. Ifa ¼ 0, the expression is equivalent to the surface segregation model.X is the regular solution interaction parameter that can be approx-imated as four times the liquid enthalpy of mixing of an equimolaralloy of the solute and solvent, DHmix. DHmix can be obtained in theliterature [54,55]. Values for the liquid enthalpy of mixing are cho-sen over those of the solid states, because such values would in-clude elastic contributions, which are considered separately in theequation. zl and zv are the coordination numbers in and out of thegrain boundary plane regions, and z is the coordination number(z ¼ zl þ 2zv ). In line with Wynblatt and Chatain [50], the close-packed planes are chosen as a surrogate model for grain boundaries,as a first order approximation. For instance, in BCC metals, theclose-packed {110} plane has the lowest surface energy and hencethe population of near {110} grain boundaries will occur to mini-mize the total energy in polycrystals at higher temperatures; exper-imental measurements of the grain boundary character distributionhas shown this to be the case [56–59]. The grain boundary param-eters could be systemically varied to understand the model uncer-tainty due to this assumption; this was not pursued within thepresent work, but has been explored previously [50]. The equationfor DHelastic

seg is based on analyses by Friedel [60] and Eshelby [61]with NAv being Avogadro’s number, KB being the bulk modulus ofthe component B; GA the shear modulus of the component A, andrA and rB are the atomic radii of components A and B. Eq. (12) cor-rectly considers both the chemical contributions (Eq. (13)) to thesegregation enthalpy, as well as the elastic enthalpy (Eq. (14)), asproposed by Wynblatt and Ku [50,53]. Recently, Saber et al. [31] ap-plied a similar model augmented with the Trelewicz and Schuh [32]analytical approach to four binary systems: Fe–Zr, Cu–Nb, Cu–Zr,Ni–W. In contrast to Saber et al. the present model uses the conceptof internal state variables to define the metastable state. Addition-ally, we apply the present model to a much larger collection of bin-ary systems (>1000) using published experimental and predictedvalues, which necessitated the development of thermodynamic sta-bility maps to organize the information and to guide alloy selection/development. The developed maps allow the visualization of trendsgoverning thermodynamic stability that are not obvious given anal-ysis of only a few systems.

3. Application to grain size stabilization in Fe–Zr and otherbinary systems

3.1. Grain boundary stabilization in the Fe–Zr system

For a given system under constant temperature, pressure, andoverall composition, the grain boundary energy represented byEq. (11) is a function of xGB

B and xIB, which are related by mass bal-

ance based on the grain size d and the grain boundary model. TheFe–Zr system is used to demonstrate the procedure. An experimen-tal investigation for the stabilization of nanocrystalline Fe by addi-tions of Zr solute was reported in Ref. [48]. For BCC Fe with {110}grain boundary planes, zl ¼ 4 and zv ¼ 2, and a ¼ 5=6. It is assumedthat rA ¼ rB ¼ r ¼ rFe ¼ NAvV2=3

Fe where VFe is the atomic volumeof Fe (solvent).

For a bilayer grain boundary model shown schematically inFig. 1, one can make the following approximation:

C ¼ @nB

@A¼

2 xGBZr � xI

Zr

� �r

: ð15Þ

where C is the grain boundary solute excess. With d being the vol-ume-averaged grain size, the mass conservation in a closed systemwith a mole fraction x0 of Zr added to Fe is

x0 ¼2Vm xGB

Zr � xIZr

� �r

3dþ xI

Zr; ð16Þ

where 3=d represents the grain boundary area per unit volume forthe spherical grain shape, and Vm is the molar volume for the sol-vent Fe. Eq. (11) can thus be re-written in terms of the normalizedgrain boundary energy c=c0.

cc0¼ 1þ

2 xGBZr � xI

Zr

� �c0r

cZr � cFe

6r� DHmix

173

xGBZr � 6xI

Zr þ16

�

þ DHelasticseg � RT ln

xIZr 1� xGB

Zr

� �1� xI

Zr

� �xGB

Zr

" #): ð17Þ

0 0.1 0.2 0.3 0.4 0.5−0.2

0

0.2

0.4

0.6

0.8

1

Zr grain boundary mole fraction, xZrGB

Nor

mal

ized

gra

in b

ound

ary

ener

gy, γ

/γ0

d = 10 nmd = 15 nmd = 23.1 nmd = 30 nmd = 50 nmd = 1e12 nm

Fig. 2. Normalized grain boundary energy c=c0 versus mol fraction of Zr atoms onthe grain boundary for several different grain sizes: 10, 15, 23.1, 30, 50 and1 � 1012 nm. The molar Zr fraction x0 is 0.03 and the temperature T is 550 �C for thisplot. The plot intersects c=c0 ¼ 0 at a grain size of 23.1 nm, which corresponds tothe stabilized grain size, dm .

Table 1Parameters of the Fe–Zr binary system.

Property Value/Units Equation or Reference

a 5/6 a ¼ �intergranular=�intragranular

z 8 BCC, z ¼ zl þ 2zvzv 2 BCC, {110}zl 4 BCC, {110}c0 0.795 J m�2 Ref. [62,63]cZr 1.909 J m�2 Ref. [62,63]cFe 2.417 J m�2 Ref. [62,63]KZr 89.8 GPa Ref. [64]GFe 81.6 GPa Ref. [64]VFe 0.0118 nm3 Ref. [64]VZr 0.0233 nm3 Ref. [64]DHmix �25 kJ mol�1 Ref. [54,55]

DHelasticseg �108 kJ mol�1 Eq. (15)

r 31217 m2 mol�1 NAv V2=3Fe

Vm 7.107 � 10�6 m3 mol�1 NAv VFe


Fig. 2 shows a typical plot of c=c0 versus xGBZr for a range of grain

size values at x0 ¼ 0:03 and T ¼ 550 �C with other parameters gi-ven in Table 1. It can be seen that each c=c0 curve passes througha minimum for each grain size considered. The equilibrium condi-tion of the system with respect to grain growth [42] is stipulatedby dG ¼ 0, which is equivalent to setting Eq. (17) equal to zero,i.e., c=c0 ¼ 0. Fig. 2 shows that the curve with d ¼ 23.1 nm (filledred1 circles) has a minimum at xGB

Zr ¼ 0.26 with c=c0 ¼ 0. This grainsize value is designated as dm. For systems with d < dm, there willbe a thermodynamic driving force for grain growth to reduce thegrain boundary area until d ¼ dm. For systems with d > dm, thereare two xGB

Zr values where c=c0 ¼ 0 as shown in Fig. 2. However, itis evident that if these two states were put together, atomic diffusionwould take place due to the different values of xGB

Zr and xIZr of the two

states until both xGBZr and xI

Zr become homogeneous in grain boundaryand grain interior, respectively, and the grain size adjusts itself to dm

based on the model.Fig. 3(a) shows the dm grain size values as a function of temper-

ature for a range of x0 alloy contents. At higher solute concentra-tions x0, there is additional solute to stabilize the grainboundaries. At the higher Zr contents, stabilization at a nanoscalegrain size smaller than 100 nm would be effective up to tempera-tures from 800 �C to 900 �C. There is an abrupt destabilization astemperature increases above a limit, though. An alternative plotof the data in Fig. 3(a) is given in Fig. 3(b) where 1=dm is plottedas a function of T and x0. Complete loss of stabilization is revealedas these curves approach the dm !1 limit. Using a regular solu-tion model in the limit of small x0 values and fully saturated grainboundaries, Kirchheim [42] suggested that inverse grain size ver-sus ln T curves would be linear with a negative slope, in qualitativeagreement with the plot in Fig. 3(a).

The grain boundary thermodynamic stabilization model used inthe present work includes elastic size misfit energy, DHelastic

seg , andchemical bond energy DHchem

seg . DHchemseg is manifested by the enthalpy

of mixing, DHmix, of an equimolar Fe–Zr liquid phase through thebond energy interaction parameter, a; however, the DHchem

seg is dom-inated by the magnitude of the enthalpy of mixing DHmix. DHelastic

seg

1 For interpretation of color in Fig. 2, the reader is referred to the web version ofthis article.

always favors grain boundary segregation. DHmix > 0 (demixing)favors grain boundary segregation whereas DHmix < 0 (mixing) fa-vors grain boundary desegregation. The combination of these twocontributions (DHmix / DHchem

seg and DHelasticseg ) dictates the effect of

an alloying element on grain boundary energy. The effect ofDHmix for Fe–Zr alloys is examined in Fig. 4 for x0 ¼ 0.04 by system-atically changing the DHmix values. For a hypothetical case of nochemical effect with DHmix ¼ 0; dm ¼ 10 nm is predicted at themelting point of Fe. In an Fe–4%Zr alloy annealed at 913 �C, trans-mission electron microscopy and ion channeling contrast imagesindicated a volume average grain size of 57 ± 15 nm [16]. In addi-tion, X-ray diffraction data on the same alloy as a function ofannealing temperature are provided and plotted as blue stars inFigs. 3 and 4. A value of DHmix ¼ �24 kJ mol�1 is in excellent agree-ment with these experimental results and is pretty close to the�25 kJ mol�1 obtained from the literature (see Table 1). It is nota-ble that the trends observed for the Fe–Zr results in reference [48]lend support to quantitative predictions for thermodynamicstabilization.

3.2. Grain boundary stabilization in other binary systems

The same approach is subsequently applied to other binary sys-tems of 44 solvents each with 52 solutes. Since there are a numberof different solvent crystal structures, the bilayer parameters inTable 1 for BCC Fe will change as a function of solvent: FCC/HCP(z ¼ 12; zl ¼ 6; zv ¼ 3), BCC (z ¼ 8; zl ¼ 4; zv ¼ 2), orthorhombic/tetragonal (z ¼ 6; zl ¼ 4; zv ¼ 1), and diamond (z ¼ 4; zl ¼ 2;zv ¼ 1). The supplemental text files contain the relevant data usedfor the present work collected from different sources [55,65,66].Additionally, a visual representation of the free energy modelwas developed using the present model for identifying many sys-tems where the grain size can be controlled after and during pro-cessing [67]. As an example, the periodic table for various solutesin an Fe solvent at a specified temperature, grain size and soluteconcentration is shown in Fig. 5. The magnified inset image showsthe key color-coded indicators for stabilization including elasticenthalpy, mixing enthalpy, solute concentration at the minimumgrain boundary energy, and the corresponding minimum grainboundary energy. The gray boxes indicate that there is insufficientdata for calculating the normalized grain boundary energy, due tolack of elastic constants, enthalpy of mixing, etc. In contrast to thepresent work, this slice through the free energy landscape does notdynamically change the solute concentration for each solute toindicate the required solute concentration to stabilize the grainstructure (i.e., zero grain boundary energy).

0 200 400 600 800 10000

20

40

60

80

100

120

Temperature, T (ºC) Temperature, T (ºC)

Stab

ilize

d gr

ain

size

, dm

(nm

)

0.0100.0150.0200.0300.0400.050XRD data

Mol Fraction Zr

0 200 400 600 800 10000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Inve

rse

stab

ilize

d gr

ain

size

, 1/d

m (

nm−1

)

0.0100.0150.0200.0300.0400.050XRD data

Mol Fraction Zr

(a) (b)

Fig. 3. (a) Stabilized grain size dm and (b) inverse grain size as a function of temperature T for different Zr molar fractions, x0: 0.01, 0.015, 0.02, 0.03, 0.04 and 0.05.Experimental XRD and TEM data for an Fe–4%Zr alloy are plotted as blue stars [16,48]. DHmix of �24 kJ mol�1 is used to show agreement with the experimental XRD/TEM data.The inverse grain size shows a linear dependence on temperature T with some deviation from linearity at lower temperatures. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)


As a first example, Table 2 lists the solute concentrations re-quired to thermally stabilize the binary system at a given grain size(25 nm) with a given solute for a particular solvent (in this case, Fe)at a given temperature (0:60TFe

M � 1200 K). For each solute (of 52),the following quantities were obtained for the solvent: the enthal-py of mixing DHmix, the free surface energy cB, the bulk modulus KB,the molar volume VB

M , and atomic radii rB. Some of these are re-quired to calculate the elastic enthalpy term DHelastic

seg using Eq.(14). Then, the content of each solute was changed from 0.1% to10% in increments of 0.1% to find the minimum solute concentra-tion required to stabilize the system, i.e., c=c0 ¼ 0 (Eq. (17)). Table 2represents all solutes and their minimum concentration x0 re-quired to stabilize the solvent Fe at a grain size of 25 nm for a tem-perature of 0:60TFe

M . The elastic enthalpy DHelasticseg and enthalpy of

mixing DHmix are also included along with the grain boundary sur-face excess for the bilayer model, which represent reasonable

0 200 400 600 800 1000 12000

10

20

30

40

50

60

70

80

90

100

Stab

ilize

d gr

ain

size

, dm

(nm

)

0−20−24−25−26−30XRD data

ΔHmix

(kJ mol−1)

Temperature, T (ºC)

Fig. 4. Stabilized grain size dm versus temperature T for molar Zr fraction x0 ¼ 0:04and different DHmix values: 0; �20; �24; �25; �26; �30 kJ mol�1. ExperimentalXRD and TEM data for an Fe–4%Zr alloy are plotted as blue stars [16,48].DHmix ¼ �24 kJ mol�1 shows excellent agreement with the experimental XRD/TEM data. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

values. This table does not include those solutes that were not pre-dicted to stabilize the system, c=c0 > 0 for all x0 6 10%. For stabil-ization of a larger grain size at the same temperature, lower soluteconcentrations are required (e.g., for 100 nm, 4.8% Zr or 0.2% Ca).This would be expected, since there is a lower total grain boundaryarea at larger grain sizes. Similarly, for stabilization of the samegrain size at a lower temperature, lower solute concentrations areagain required. While there are a number of solutes that wouldnot be selected due to their high price, toxicity, low melting tem-perature, etc., this model may shed light on prospective binarynanocrystalline systems from a thermodynamic stability stand-point. Additionally, some solutes may drive the interfacial energydown, but could embrittle the grain boundary. Hence, this modelis only one criteria of several that would be important for selectingnanocrystalline binary systems.

The solutes that are effective at stabilizing the nanocrystallinegrain structure are functions of both the elastic enthalpy and theenthalpy of mixing. Fig. 6 is a plot of the elastic enthalpy versusthe enthalpy of mixing for both the stabilizing solutes (Table 2)as well as the solutes that do not appear in Table 2. The red andblack dots denote the stabilizing and non-stabilizing solutes, andthe size of the dot for the stabilizing solutes corresponds to themagnitude of the minimum solute concentration required to stabi-lize a grain size of 25 nm. Moreover, a convex hull of all of thesystems (dotted line), the stabilizing solutes (red line), and thenon-stabilizing solutes (black line) is plotted to delineate the nano-crystalline stability design space. In computational geometry, aconvex hull for of a set X of points is defined as the smallest convexset that contains X. Interestingly, there is a noticeable division ofthe solutes that stabilize the nanocrystalline grain structure fromthose that do not. Therefore, Fig. 6 and similar plots can be envi-sioned as a map of nanocrystalline stability.

There are a few general trends that emerge from the nanocrys-talline stability map (Fig. 6). The first general trend observed is thatthe larger the magnitude of the elastic enthalpy (i.e., more nega-tive) of the solute, the smaller the bulk solute concentration re-quired for stabilization. Additionally, the magnitude and sign ofthe enthalpy of mixing detract from or reinforce stabilization, withnegative values tending to reduce stability in any given system.Furthermore, stabilized systems that require higher solute concen-trations tend to have a lower driving force for segregating to theboundary. Of the non-stabilized systems, there were a number ofsystems that reduced the excess grain boundary free energy, but

Fig. 5. Visual representation of the model given fixed temperature, bulk solute concentration, and grain size.

Table 2Solutes and their concentrations that stabilize the grain structure in nanocrystallineFe at temperature T ¼ 0:60TFe

M ¼ 921 �C (1194 K) and grain size dm ¼ 25 nm.

System DHelasticseg

DHmix x0 C

(Fe–X) (kJ mol�1) (kJ mol�1) (at.%) (lmol m�2)

Th �187 �11 0.6 6.4Ca �143 25 0.7 7.6Pb �135 29 0.7 7.6Bi �135 26 0.7 7.6Sr �123 34 0.8 8.7Y �149 �1 0.8 8.5Ba �120 37 0.8 8.7La �138 5 0.8 8.5K �50 81 0.9 9.9Rb �53 83 0.9 9.9Sn �123 11 0.9 9.5Sb �124 10 0.9 9.5Cs �46 85 0.9 9.9Tl �77 31 1.1 11.7Na �36 62 1.2 13.1In �73 19 1.4 14.4Mg �52 18 1.9 19.1Hg �45 22 1.9 19.5Ag �31 28 2.0 20.8Cd �47 17 2.1 21.0Li �14 26 2.5 26.0Sc �95 �11 2.7 18.4Au �39 8 4.4 30.9Hf �106 �21 5.0 17.1Zr �109 �25 5.8 16.1

−250 −200 −150 −100 −50 0 50−150

−100

−50

0

50

100

150

Elastic enthalpy, ΔH elasticseg (kJ mol−1)

Ent

halp

y of

mix

ing,

ΔH

mix

(kJ

mol

−1)

Fe

Black dots did not stabilize for x0 < 10% Red dots denote stabilized binary systemsMarker size denotes stabilized bulk solute concentration, x0

Minimum: 0.6% (Th)Intermediate: 2.7% (Sc)Maximum: 5.8% (Zr)

Fig. 6. Nanocrystalline Fe stability map that plots the elastic enthalpy versus theenthalpy of mixing for both the stabilizing solutes (Table 2, plotted as red dots) aswell as the solutes that do not appear in Table 2, plotted as black dots. The size ofthe dot for the stabilizing solutes corresponds to the magnitude of the minimumsolute concentration required to stabilize a grain size of 25 nm.


did not reduce it to zero (recall that c=c0 ¼ 0 is required for stabi-lized systems); in a very limited number of cases, bulk solute con-centrations greater than 10% reduced the grain boundary energy tozero. However, in general, increasing the solute content even tomoderately high levels (>5%) is not always a good practice as pre-cipitate formation is exacerbated at higher solute contents andwith increased temperature. Experimental observations generallyshow that secondary phases must be kinetically hindered [39]from forming, since they compete for the available solute, causingdestabilization of the boundaries and rapid grain coarsening. Addi-tionally, attention must be paid to the extent of forced solubilitythat can be attained by a given processing method, as this ulti-mately limits the amount of solute available for segregation toboundaries.

Using the above rational, tables for other solvents that are oftenassociated with nanocrystalline materials are also presented. The

other solvents are Ni, Cu, Al, Mg, Ti, Pd, and W (Appendix A, TablesA.3, A.4, A.5, A.6, A.7, A.8 and A.9). Moreover, the same nanocrystal-line stability maps are shown for the stabilizing solutes in thesesystems as well (Figs. 7–13). A few general themes are observedfor these systems:

1. In a number of systems, the majority of stabilizing solutes havea positive enthalpy of mixing. For instance, the majority ofstabilizing solutes in the Mg, Ti, Al, and W systems have apositive enthalpy of mixing. Currently the authors are una-ware of any thermodynamic reports of grain growth pre-vention for nanocrystalline Mg, Ti, Al, or W. Additionally,it should be noted that many of the predicted stabilizingagents (e.g., Cs, Rb, K, Ba, Na, In, Sn, Tl, Sb and Pb), have beensuggested by Seah to be grain boundary embrittling agents

−250 −200 −150 −100 −50 0 50−150

−100

−50

0

50

100

150

Elastic enthalpy, ΔH elasticseg

(kJ mol−1)

Ent

halp

y of

mix

ing,

ΔH

mix

(kJ

mol

−1)

Ni

Black dots did not stabilize for x0 < 10%

Red dots denote stabilized binary systemsMarker size denotes stabilized bulk solute concentration, x

0

Minimum: 0.6% (Th)Intermediate: 3.4% (Cd)Maximum: 6.1% (Mg)

Fig. 7. Nanocrystalline Ni stability map that plots the elastic enthalpy versus theenthalpy of mixing for both the stabilizing solutes (Table A.3, plotted as red dots) aswell as the solutes that do not appear in Table A.3, plotted as black dots. The size ofthe dot for the stabilizing solutes corresponds to the magnitude of the minimumsolute concentration required to stabilize a grain size of 25 nm. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

−250 −200 −150 −100 −50 0 50−150

−100

−50

0

50

150

100

Elastic enthalpy, ΔH elastic

seg (kJ mol−1)

Ent

halp

y of

mix

ing,

ΔH

mix

(kJ

mol

−1)

Cu



0

Minimum: 0.5% (Th)Intermediate: 4.6% (Hf)Maximum: 8.4% (Ta)

Fig. 8. Nanocrystalline Cu stability map that plots the elastic enthalpy versus theenthalpy of mixing for both the stabilizing solutes (Table A.4, plotted as red dots) aswell as the solutes that do not appear in Table A.4, plotted as black dots. The size ofthe dot for the stabilizing solutes corresponds to the magnitude of the minimumsolute concentration required to stabilize a grain size of 25 nm. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

−250 −200 −150 −100 −50 0 50−150

−100

−50

0

50

100

150


Ent

halp

y of

mix

ing,

ΔH

mix

(kJ

mol

−1)

Al



0

Minimum: 1% (Bi)Intermediate: 2.2% (Na)Maximum: 3.9% (B)

ig. 9. Nanocrystalline Al stability map that plots the elastic enthalpy versus thenthalpy of mixing for both the stabilizing solutes (Table A.5, plotted as red dots) asell as the solutes that do not appear in Table A.5, plotted as black dots. The size ofe dot for the stabilizing solutes corresponds to the magnitude of the minimum

olute concentration required to stabilize a grain size of 25 nm. (For interpretationf the references to color in this figure legend, the reader is referred to the webersion of this article.)

−250 −200 −150 −100 −50 0 50−150

−100

−50

0

50

100

150


Ent

halp

y of

mix

ing,

ΔH

mix

(kJ

mol

−1)

Mg



0

Minimum: 1.2% (Cs)Intermediate: 3% (Re)Maximum: 5% (Co)

Fig. 10. Nanocrystalline Mg stability map that plots the elastic enthalpy versus theenthalpy of mixing for both the stabilizing solutes (Table A.6, plotted as red dots) aswell as the solutes that do not appear in Table A.6, plotted as black dots. The size ofthe dot for the stabilizing solutes corresponds to the magnitude of the minimumsolute concentration required to stabilize a grain size of 25 nm. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)


[68]. Interestingly, in Ti, all of the stabilizing solutes have apositive enthalpy of mixing and there is a clear delineationbetween the stabilizing and non-stabilizing solutes. In theW system, a large variety of possible stabilizing solutes

Fewthsov

exist with the majority of the systems requiring a largerbulk solute content than many of the other systems exam-ined. Additionally, due to the high melting point of W, hightemperature stability will suffer from intermetallic forma-tion at much lower temperatures.

−250 −200 −150 −100 −50 0 50−150

−100

−50

0

50

100

150


Ent

halp

y of

mix

ing,

ΔH

mix

(kJ

mol

−1)

Ti



0

Minimum: 1% (Ba)Intermediate: 2.8% (Li)Maximum: 8.8% (Mg)

Fig. 11. Nanocrystalline Ti stability map that plots the elastic enthalpy versus theenthalpy of mixing for both the stabilizing solutes (Table A.7, plotted as red dots) aswell as the solutes that do not appear in Table A.7, plotted as black dots. The size ofthe dot for the stabilizing solutes corresponds to the magnitude of the minimumsolute concentration required to stabilize a grain size of 25 nm. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

−250 −200 −150 −100 −50 0 50−150

−100

−50

0

50

100

150

Elastic enthalpy, ΔH elasticseg

(kJ mol−1)

Ent

halp

y of

mix

ing,

ΔH

mix

(kJ

mol

−1)

Pd


Red dots denote stabilized binary systems

Marker size denotes stabilized bulk solute concentration, x0

Value: 6.6% (Bi)

Fig. 12. Nanocrystalline Pd stability map that plots the elastic enthalpy versus theenthalpy of mixing for both the stabilizing solutes (Table A.8, plotted as red dots) aswell as the solutes that do not appear in Table A.8, plotted as black dots. The size ofthe dot for the stabilizing solutes corresponds to the magnitude of the minimumsolute concentration required to stabilize a grain size of 25 nm. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

−250 −200 −150 −100 −50 0 50−150

−100

−50

0

50

100

150

Ent

halp

y of

mix

ing,

ΔH

mix

(kJ

mol

−1)

W

Minimum: 1.2% (Rb)Intermediate: 3.6% (Ag)Maximum: 5.9% (Be)



0


Fig. 13. Nanocrystalline W stability map that plots the elastic enthalpy versus theenthalpy of mixing for both the stabilizing solutes (Table A.9, plotted as red dots) aswell as the solutes that do not appear in Table A.9, plotted as black dots. The size ofthe dot for the stabilizing solutes corresponds to the magnitude of the minimumsolute concentration required to stabilize a grain size of 25 nm. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)


2. A positive enthalpy of mixing is not necessarily required fornanocrystalline stability. For instance, in the Ni- and Cu-based stabilized systems, an equal number of positive andnegative enthalpy of mixing systems are indicated over

the given range. Even for Fe, Al, Mg, W, there are severalstabilizing solutes with negative enthalpies of mixing.However, the stabilizing solutes all tend to have a greatermagnitude of elastic enthalpy in these cases to balanceout the negative enthalpy of mixing. On the other hand,the majority of the non-stabilized systems have a negativeenthalpy of mixing values and tend to form intermetallics.This is as expected, since negative enthalpy of mixing val-ues lower the segregation enthalpy and the system ordersresulting in oscillations of the composition; for positiveenthalpy of mixing values, segregation is enhanced byforming clusters and/or by multi-layer segregation [69].

3. The limited experimental data supports the grouping of stabi-lizing and non-stabilizing solutes from the nanocrystalline sta-bility maps. Experimentally, there are few studied systemsfor either Ni or Cu, for which the grain growth was classi-fied as being controlled thermodynamically. While a lackof grain growth was observed for nanocrystalline Ni–Wand it is reasoned to be thermodynamic in nature, esti-mates by Schuh et al. suggested only a 60% reduction inexcess grain boundary energy was possible, and thus trueequilibrium was not met [20]. Large reductions in grainboundary energy were calculated for W solute in Ni withthe present model (�40–60% at high solute concentration>10%); since true thermodynamic equilibrium was notreached, W is not suggested to be a true stabilizing agentfor Ni (Table A.3). Additionally, Cu has been experimentallyadded to nanocrystalline Ni and has shown minor improve-ments in delaying the onset of grain growth at T < 0:33Tm

[70]; however, in agreement with our stability maps, Cu isnot listed as a stabilizing solute for annealing temperaturesT < 0:50Tm.There have been a few studies which examined thermody-namically stabilizing nanocrystalline Cu with elements


such as Ta and Zr, which are both listed as potential candi-dates (Table A.4). Recent atomic simulations on Ta dis-solved in nanocrystalline Cu have shown that in thesegregated state, Ta can prevent grain growth even at highhomologous temperatures >0.70TCu

M [24]. Zr additions to Cuhave shown mixed support of thermodynamic stability; thedelineation between kinetic and thermodynamic modes ofstability were unclear [13]. The data in Table A.4 indicates ahigh Zr concentration needed for stabilization. Interest-ingly, based on experimental results of several binary sys-tems presented in the nanocrystalline stability maps,most binary systems that are effective at grain stabilizationtend to lie at the boundary of the convex hull for stabilizingsolutes (Fe–Zr, Cu–Nb), and these are close to those solutesthat are not predicted to be effective grain stabilizers. Onthe other hand, those systems that have not been effectiveat grain stabilization (Cu–Zr) tend to lie on the interior ofthe convex hull.

4. In some systems, most solutes tend to have a negativeenthalpy of mixing, thus limiting the thermodynamic stabilityof the nanocrystalline grain structure. For example, in the Pdsystem, there is only one solute (Bi) that can stabilize thegrain size. However, it is well known that Bi acts as anembrittlement agent at grain boundaries. Since most sol-

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

12 ML

1 ML

0.5 ML

0.25 ML

Bulk Concentration (x0)

Gra

in B

ound

ary

Con

cent

rati

on (

xGB

B)

(a) (

Fig. 14. (a) Grain boundary solute concentration xGBB and (b) grain interior solute concentr

the nanocrystalline grain size at a grain size of 25 nm and a temperature of 0:6TAM (TA

M i

0 0.02 0.04 0.06 0.08 0.1100

101

102

103

104


Gra

in B

ound

ary−

Inte

rior

Con

cent

rati

on R

atio

(xG

BB

/xI B)

(a)Fig. 15. The ratio of the solute concentration in the grain boundary to the grain interior asa temperature of 0:6TA

M for (a) a stabilized grain size of 25 nm and (b) stabilized grain s

utes for Pd have only negative enthalpies of mixing, thismay help explain the lack of stabilizing solutes in thisnanocrystalline system.

3.3. Relationship between solute concentrations

The relationship between the bulk solute concentration, thegrain boundary solute concentration and the grain interior soluteconcentration for all stabilized solvent–solute combinations (566stabilized systems total from 1294 total systems, 44%) is discussedherein. There are a few trends that appear from examining theserelationships. As the bulk solute concentration required to stabilizethe nanocrystalline grain structure increases, the concentration inthe grain boundary increases as well (Fig. 14). In Fig. 14(a), noticethat the grain boundary concentration increases as a function ofthis minimum bulk solute concentration. Since there is a limitedamount of grain boundary area, this trend suggests that there isa limit to the amount of bulk solute concentration that can beadded to stabilize some systems through this mechanism. Themonolayer (ML) amounts are also plotted. Since a bilayer is as-sumed in the present model, a grain boundary concentration of100% would indicate complete coverage of two layers at the grainboundary. Most of the stabilized systems have bulk concentrationsbelow 10% and less than 2 ML (558 of 566 systems); of those, the


0 0.02 0.04 0.06 0.08 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

BG

rain

Int

erio

r C

once

ntra

tion

(x

I)

b)

ation xIB as a function of the minimum bulk solute concentration required to stabilize

s the melting temperature of the corresponding solvent A).


Gra

in B

ound

ary−

Inte

rior

Con

cent

rati

on R

atio

(xG

BB

/xI B)

0 0.02 0.04 0.06 0.08 0.1100

101

102

103

104

10 nm 25 nm 50 nm100 nm

Grain Size

(b)a function of the bulk solute concentration for stabilized nanocrystalline systems at

izes of 10, 25, 50 and 100 nm.


present model predicts that 22% of the systems stabilize with lessthan 0.25 ML, 62% stabilize with less than 0.5 ML, and 90% stabilizewith less than 1 ML at the grain boundaries. Furthermore, the ver-tical lines in Fig. 14(a) indicate the minimum bulk solute concen-tration to achieve {0.25, 0.5, 1} ML coverage, which is as follows:0.7% for 0.25 ML, 1.6% for 0.5 ML, and 3.3% for 1 ML. An alternativeway to interpret this is that the present model predicts that, for in-stance, bulk solute concentrations less than 3.3% will have lessthan 1 ML coverage at the grain boundaries. These values and plotsare for a given grain size of 25 nm and at a temperature of 0.6Tm;these values will change with both grain size and temperature.

The relationship between the interior concentration and thebulk concentration using the present model may also be of interest.In Fig. 14(b), the grain interior solute concentration deviates fromthe 1:1 line drawn as solute segregates from the grain interior tothe grain boundary, effectively resulting in a lower grain interiorconcentration. As the stabilized grain size increases, and grainboundary volume decreases, the solute concentration in the graininterior will approach the bulk solute concentration, i.e., less bulksolute concentration is required to stabilize the grain size. In theaforementioned plots and analysis, there are a few systems thatappear to be outliers from the general trends. For instance, thetwo systems above the 1:1 line in Fig. 14(b) are the K–Pt and Cs–Pt systems. In Fig. 15, there were multiple systems that were re-moved as outliers: K–Pt, Cs–W, Cs–Re, Cs–Os, Cs–Ir, Cs–Pt. Whileit is not clear why these are outliers, the ability to experimentallymeasure thermodynamic data in these systems may beproblematic.

The ratio between the solute concentration in the grain bound-ary and the grain interior changes as a function of the bulk soluteconcentration. Fig. 15(a) plots the bulk solute concentrationagainst the ratio of solute in the grain boundary to the grain inte-rior for the stabilized systems at a grain size of 25 nm and a tem-perature of 0:6TA

M (TAM is the melting temperature of the

corresponding solvent A). As the bulk solute concentration to sta-bilize the nanocrystalline grain structure increases, the ratio of sol-ute concentration in the grain boundary to the grain interiordecreases (Fig. 15). The solute concentration ratio axis in Fig. 15is plotted on a log scale to highlight the large difference in this ratioas a function of bulk solute concentration. This sort of trend is asexpected, i.e., the systems that stabilize at lower bulk solute con-centrations do so because of a greater driving force for solute tosegregate to the grain boundary and, hence, possess a higher ratioof solute in the boundary than the grain interior. Also, notice thatthe solute concentration ratio in this model is >10–100 times high-er in the grain boundary than in the grain interior for most sys-tems, which may dramatically impact properties. Fig. 15(b)shows these same systems for several different stabilized grainsizes. First, as the stabilized grain size increases, the ratio of soluteconcentration in the grain boundary to the grain interior increasesdramatically for those systems that require a low bulk solute con-centration for stabilization. This result has to do with the rapid de-crease in grain boundary volume with increasing grain sizecombined with a reduced amount of bulk solute required to stabi-lize the grain structures, which is more pronounced in the dilutelimit. By comparison, at larger bulk solute concentrations, thereis a much smaller difference in this ratio as a function of the stabi-lized grain size.

4. Summary and conclusions

A fundamental thermodynamic analysis is performed for thechange in grain boundary energy that results from solute segrega-tion to grain boundaries in a closed system. When the grain bound-ary energy is reduced to zero, the grain size can be stabilizedagainst further change. The theoretical results provide a detailed

picture of the stabilization effects, which have been validated usingpreviously-reported experiments. The developed model and stabil-ity maps aid in selecting binary systems that will open new re-gimes in processing, manufacturing and applications space.Specific findings of the present work include:

1. A methodology and model for thermodynamic stabilizationof nanocrystallinity is presented. This thermodynamic-based model allows for the prediction of a large numberof individual binary systems.

2. Numerical calculations were made for the Fe–Zr system,demonstrating that thermodynamic stabilization for nano-scale grain sizes can be expected in Fe–Zr alloys up to900 �C for Zr contents on the order of 4 at.% Zr (Fig. 3 and 4).The present model was in excellent agreement with theXRD/TEM grain size data using previously-reported soluteconcentration and enthalpy of mixing values for Fe–Zr,thereby validating this approach. Additionally, the presentmodel was used to interrogate the effect of grain size, bulksolute concentration, and grain boundary solute concentra-tion on the normalized grain boundary energy as well theeffect of bulk solute concentration, enthalpy of mixing, andtemperature on the stabilized grain size of the nanocrystal-line structure.

3. A thermodynamic stability map was made for Fe-basedsystems. These maps extend previous findings by dynami-cally incorporating how solute concentration affects grainboundary energy and stabilized grain size for a large num-ber of solutes. A strong effect of enthalpy of mixing isrevealed in the calculations (Fig. 6). We conclude that inappropriate alloy systems with suitable values of elasticand mixing enthalpy of segregation, thermodynamic stabil-ization of a nanoscale grain size is possible up to highhomologous temperatures.

4. The utility and universality of this model are demonstratedby applying it to over 1000 binary systems. Herein, weapplied this to Fe, Ni, Cu, Al, Mg, Ti, Pd and W (Tables 2,A.3, A.4, A.5, A.6, A.7, A.8 and A.9) However, these mapsmust be balanced with other properties, e.g., secondaryphase formation, solubility limit, and grain boundaryembrittlement, which must be considered as well.

Acknowledgments

This work is supported in part by the U.S. Army Research Labo-ratory (ARL) under contract GS04T09DBC0017. Dr. Mark Tschoppwould like to acknowledge partial support from ARL administeredby the Oak Ridge Institute for Science and Education through aninteragency agreement between the U.S. Department of Energyand ARL. Dr. Mark Tschopp would like to acknowledge continuedsupport for this work from ARL through the High PerformanceTechnology Group at Dynamic Research Corporation. Dr. Zi-KuiLiu would like to acknowledge support from the National ScienceFoundation under Grant DMR-1005677.

Appendix A. Supplemental: Thermodynamic stability tables

See Tables A.3, A.4, A.5, A.6, A.7, A.8 and A.9.

Appendix B. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.commatsci.2013.10.018.



Table A.3Solutes and their concentrations that stabilize the grain structure in nanocrystallineNi at temperature T ¼ 0:60TNi


System DHel DHmix x0 C(Ni–X) (kJ mol�1) (kJ mol�1) (at.%) (lmol m�2)

Th �202 �39 0.6 6.9Ca �149 �7 0.7 8.0Pb �148 13 0.7 8.2Bi �145 10 0.7 8.1Y �161 �31 0.8 8.8Sn �137 �4 0.8 9.0Sb �136 �1 0.8 9.0Sr �128 �1 0.9 10.1Ba �124 0 0.9 10.1La �147 �27 1.0 10.5K �51 45 1.1 12.7Rb �54 47 1.1 12.8Cs �46 48 1.1 12.7Tl �85 13 1.2 13.2Na �38 32 1.5 17.0In �82 2 1.7 17.4Ag �41 15 2.4 25.6Hg �51 8 2.5 25.0Au �52 7 2.9 28.0Cd �55 2 3.4 28.1Mg �60 �4 6.1 27.4

Table A.4Solutes and their concentrations that stabilize the grain structure in nanocrystallineCu at temperature T ¼ 0:60TCu


System DHelasticseg

DHmix x0 C

(Cu–X) (kJ mol�1) (kJ mol�1) (at.%) (lmol m�2)

Th �169 �24 0.5 5.4Ca �139 �13 0.6 6.4Pb �123 15 0.6 6.5Bi �127 15 0.6 6.6Y �137 �22 0.7 7.3Sb �114 7 0.7 7.5Sr �121 �9 0.8 8.4Sn �109 7 0.8 8.5Ba �118 �9 0.8 8.3La �131 �21 0.8 8.2Rb �53 27 1.1 11.7Tl �72 15 1.1 11.5K �50 25 1.2 12.7Cs �46 28 1.2 12.7In �67 10 1.3 13.2Na �36 16 1.9 19.4Hg �42 8 2.3 21.9W �20 22 2.5 26.1Cd �42 6 2.7 24.6Mo �18 19 2.8 29.0Re �14 18 3.4 31.6Hf �85 �17 4.6 16.3Zr �91 �23 6.0 14.8Mg �47 �3 6.8 26.3U �62 �7 7.0 22.2Nb �40 3 7.2 29.0Ta �44 2 8.4 28.2

Table A.5Solutes and their concentrations that stabilize the grain structure in nanocrystallineAl at temperature T ¼ 0:60TAl


System DHelasticseg

DHmix x0 C

(Al–X) (kJ mol�1) (kJ mol�1) (at.%) (lmol m�2)

Ca �97 �20 1.0 7.1Bi �66 10 1.0 7.6Rb �46 25 1.1 8.5Sr �91 �18 1.1 7.7Cs �41 26 1.1 8.4Ba �93 �20 1.1 7.7K �42 23 1.2 9.2Pb �50 10 1.4 10.3Sb �47 2 2.0 13.6Tl �29 11 2.1 15.2Na �23 13 2.2 16.1Sn �35 4 2.8 18.5In �22 7 3.4 23.3B �47 0 3.9 20.7

Table A.6Solutes and their concentrations that stabilize the grain structure in nanocrystallineMg at temperature T ¼ 0:60TMg


System DHelasticseg

DHmix x0 C

(Mg–X) (kJ mol�1) (kJ mol�1) (at.%) (lmol m�2)

Rb �38 23 1.2 6.5Cs �36 25 1.2 6.5Ba �64 �4 1.4 7.0K �32 20 1.5 8.1Cr �35 24 1.7 9.2Sr �57 �4 1.8 8.4B �70 �4 1.9 8.8Mo �16 36 1.9 10.5W �16 38 1.9 10.6Fe �36 18 2.0 10.6Nb �7 32 2.3 12.8V �20 23 2.4 13.0Ca �54 �6 2.5 10.0Ta �7 30 2.5 13.8Be �62 �3 2.6 10.8Mn �33 10 2.6 13.1Re �19 21 3.0 16.3Na �11 10 3.6 18.2Ti �8 16 3.7 19.8Co �41 3 5.0 19.7

Table A.7Solutes and their concentrations that stabilize the grain structure in nanocrystallineTi at temperature T ¼ 0:60TTi


System DHelasticseg

DHmix x0 C

(Ti–X) (kJ mol�1) (kJ mol�1) (at.%) (lmol m�2)

Rb �45 100 1.0 7.4Sr �89 53 1.0 7.3Cs �41 104 1.0 7.4Ba �91 57 1.0 7.3K �41 94 1.1 8.1Ca �96 43 1.1 7.9Na �22 68 1.5 11.0La �73 20 1.8 12.3Y �65 15 2.2 14.6Th �77 8 2.3 14.7Li �2 34 2.8 19.8Mg �10 16 8.8 21.5


Table A.8Solutes and their concentrations that stabilize the grain structure in nanocrystallinePd at temperature T ¼ 0:60TPd


System DHelasticseg

DHmix x0 C

(Pd–X) (kJ mol�1) (kJ mol�1) (at.%) (lmol m�2)

Bi �93 �21 6.6 14.6

Table A.9Solutes and their concentrations that stabilize the grain structure in nanocrystallineW at temperature T ¼ 0:60TW


System DHelasticseg

DHmix x0 C

(W–X) (kJ mol�1) (kJ mol�1) (at.%) (lmol m�2)

Rb �48 129 1.2 9.7K �44 124 1.3 10.6Cs �42 132 1.3 10.6Ba �102 74 1.3 10.3Ca �115 57 1.4 11.0Sr �103 70 1.4 11.1Na �26 97 1.6 12.8Pb �87 49 1.8 13.8Bi �96 45 1.8 13.8La �101 32 2.0 14.8Th �129 12 2.2 15.2Y �102 24 2.3 16.6Tl �46 52 2.3 17.8Sb �80 25 2.7 19.3Sn �71 27 2.8 20.2In �40 38 2.9 21.7Li �5 50 3.0 23.2Hg �22 38 3.3 24.7Mg �23 38 3.4 25.6Ag �2 43 3.6 26.1Cd �17 33 4.1 25.8B �173 �31 5.0 14.3Sc �49 19 5.4 25.1Be �113 �3 5.9 22.1


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